The Tutte polynomial of a graph, depth-rst. search, and simplicial complex partitions

Transcription

1 The Tutte polynomial of a gaph, depth-st seach, and simplicial complex patitions Dedicated to Dominique Foata on the occasion of his 60th bithday Ia M. Gessel Depatment of Mathematics Bandeis Univesity Waltham, MA Buce E. Sagan Depatment of Mathematics Michigan State Univesity East Lansing, MI Submitted: Octobe 18, 1994; Accepted: June 14, 1995 Key Wods: Tutte polynomial, simplicial complex, patition, depth-st seach, spanning tee, acyclic oientation, invesion, paking function AMS subject classication (1985): Pimay 05C30; Seconday 05C05, 68R10.

2 Abstact One of the most impotant numeical quantities that can be computed fom a gaph G is the two-vaiable Tutte polynomial. Specializations of the Tutte polynomial count vaious objects associated with G, e.g., subgaphs, spanning tees, acyclic oientations, invesions and paking functions. We show that by patitioning cetain simplicial complexes elated to G into intevals, one can povide combinatoial demonstations of these esults. One of the pimay tools fo poviding such a patition is depth-st seach.

3 1 Intoduction and denitions Tutte dened the polynomial that beas his name as the geneating function fo two paametes, namely the intenal and extenal activities, associated with the spanning tees of a connected gaph. In this pape we will popose a numbe of new, but elated, notions of extenal activity. We will show in the next six sections that using these othe denitions of activity can lead to combinatoial poofs of esults about many specializations of t G. These evaluations count subgaphs, acyclic oientations, subdigaphs, invesions and paking functions. The basic idea is to use depth-st seach to associate a spanning foest F with each object to be counted. This patitions the simplicial complex of all objects (odeed by inclusion) into intevals, one fo each F. Evey inteval tuns out to be a Boolean algeba consisting of all ways to add extenally active edges to F. Expessing the Tutte polynomial in tems of sums ove such intevals pemits us to extact the necessay combinatoial infomation. The idea of using patitions to get infomation about the Tutte polynomial goes back to Capo [8] and has also been used by Bai [2], Dawson [9, 10], Gessel and Wang [18], Godon and Taldi [20], and othes. See Bjone [4] fo a good account of the connection between Tutte polynomials, patitions, and shellability. See Bylawski [5] and his suvey with Oxley [6] fo the geneal theoy of the Tutte polynomial. Patitioning simplicial complexes into Boolean algebas also has othe applications. See, fo example, the pape of Gasia and Stanton [15]. Finally, we should mention that Kleitman and Winston [23] have used depth-st seach to constuct a bijection in a context simila to ous. Howeve, ou pape is the st to systematically mine the combination of the patitioning and DFS ideas to obtain a wide ange of esults. Let G denote a gaph with vetex set V. In most of ou wok (in paticula, fo depth-st seach and its vaiations) we assume that V is totally odeed. Often we will take V = f1; 2; : : : ; ng. We will pemit G's edge set to have loops and multiple edges, calling two edges with the same endpoints paallel. It will be convenient to identify a gaph with its edge set and use the notation G fo both. All the pevious conventions will apply to digaphs as well. All of ou subgaphs ae spanning, that is, a subgaph of G has the same vetex set as G. Thus we identify subgaphs of G with subsets of the edge set. A subgaph of G is a foest if it is acyclic; in paticula, it must contain no loops o paallel edges. The connected components of a foest ae tees. Befoe dening the Tutte polynomial, t G (x; y), we make the convention that, except whee stated othewise, G is a connected gaph. This is no loss of geneality fo two easons. Fist, if G is disconnected then t G (x; y) is just the poduct of the Tutte polynomials of the components of G. Futhemoe, most of the algoith-

4 the electonic jounal of combinatoics 3 (2) (1996), #R9 2 mic constuctions that we will use can be caied out in G by applying them to each component. In the few places whee this is not tue, we will indicate what modications need to be made fo the geneal case. Now suppose we ae given G and a total odeing of its edges. Conside a spanning tee T of G. An edge e 2 G? T is extenally active if it is the lagest edge in the unique cycle contained in T [ e. We let and EA(T ) = set of extenally active edges of T ea(t ) = jea(t )j whee j j denotes cadinality. Of couse, the set of extenally active edges depends on both G and T. Howeve, G will always be clea fom context. An edge e 2 T is intenally active if it is the lagest edge in the unique cocycle contained in (G? T ) [ e. (A cocycle is a minimal disconnecting subset of G.) We let and IA(T ) = set of intenally active edges of T ia(t ) = jia(t )j: Tutte [40] then dened his polynomial as t G (x; y) = T G x ia(t ) y ea(t ) (1) whee the sum is ove all spanning tees T of G. Tutte showed that t G is welldened, i.e., independent of the odeing of the edges of G. Hencefoth, we will not assume that the edges of G ae odeed unless it is explicitly stated. We end this section by eviewing the notion of depth-st seach, which we abbeviate to DFS. Given a gaph H with vetex set V, we will use the following algoithm to ceate the DFS foest F of H. DFS1 Let F := ;. DFS2 Let v be the least unvisited vetex in V. Mak v as visited. DFS3 Pick some unvisited vetex u adjacent to v by an edge e if such a vetex exists. Mak u as visited and set v := u; F := F [ e. Repeat this step until v has no unvisited neighbos. DFS4 If thee is a visited vetex with unvisited neighbos, let v be the most ecently visited such vetex and go to DFS3. Othewise, go to DFS2 and epeat the pocess until all vetices ae visited. The DFS vaiants that we will intoduce in the next sections ae all constucted by specifying the vetex u and the edge e in step DFS3.

5 the electonic jounal of combinatoics 3 (2) (1996), #R9 3 2 Subgaphs Let G be a gaph with jv j = n, and let H be a subgaph. We denote by c(h) the numbe of components of H. We intoduce two useful invaiants associated with H, namely (H) = c(h)? c(g) = c(h)? 1 (2) and (H) = jhj? jv j + c(h) = jhj? n + c(h) (3) whee we ecall that jhj denotes the numbe of edges in H. These quantities ae natually associated with the cycle matoid of G whose independent sets consist of all spanning foests F of G. The ank of H G in this matoid is and the coank is (H) = maxfjf j : F H with F a spanning foest of Gg = jv j? c(h) (H) = maxfjcj : C H with G? C connectedg = jhj? c(g? H) + 1: Thus and (H) = (G)? (H) (H) = (G)? (G? H): It is well known [4] that one evaluation of the Tutte polynomial is the geneating function fo subgaphs of G with espect to the invaiants and. Theoem 2.1 We have t G (1 + x; 1 + y) = HG x (H) y (H) (4) whee the sum is ove all subgaphs H of G. The vesion of depth-st seach that will be useful in connection with subgaphs H G is geatest-neighbo DFS. Each time we pefom DFS3 we visit the unvisited neighbo with lagest label st. (Howeve, we still always stat the seach at the vetex with smallest label. One of the easons fo these conventions is to coincide with those used late when discussing invesions in tees.) We also assume that each set of paallel edges is totally odeed and that we always take

6 the electonic jounal of combinatoics 3 (2) (1996), #R ? 4? B B BB 8 a A A BB 2 5 d 3 1? 4? 8 a A A @ G H G F = F(H) Figue 1: A geatest-neighbo DFS foest F fo H G the lagest edge connecting two vetices. If H has geatest-neighbo DFS foest F, then we wite F + (H) = F which we will often abbeviate to F(H) = F. It is convenient to oot each tee of F(H) at its least vetex. We also note that F(H) does not depend on the edges of G? H and that c(f(h)) = c(h). By way of illustation, suppose we have the gaph G in Figue 1. The two sets of paallel edges ae odeed lexicogaphically. Then the given subgaph H has geatest-neighbo foest F with components ooted at vetices 1 and 3. Now given a spanning foest F G let us say that an edge e 2 G? F is (geatest-neighbo) extenally active if F(F [ e) = F: We wite E + (F ), o simply E(F ), fo the set of geatest-neighbo extenally active edges. In ou pevious example, edges f1; 2g; f3; 7g; f2; 2g and a = f4; 6g ae extenally active while f1; 5g; f5; 7g and d = f7; 8g ae not. The next esult follows easily fom the denitions. In it, ] stands fo disjoint union. Poposition 2.2 If H is any subgaph and F is any spanning foest of G then F(H) = F () F H F ] E(F ): Thus the intevals [F; F ] E(F )] patition the simplicial complex of all subgaphs of G into Boolean algebas, one fo each spanning foest. To tun this poposition into an enumeative esult, note that if F(H) = F then c(h) = c(f ) so (H) = (F ) = c(f )?1 and (H) = jhj?jf j = jh \E(F )j. Thus, if we x a foest F and sum ove the coesponding inteval x (H) y (H) = x (F ) H:F(H)=F AE(F ) y jaj

7 the electonic jounal of combinatoics 3 (2) (1996), #R9 5 Summing ove all foests F, we have HG x (H) y (H) = F G = x (F ) (1 + y) je(f )j : x (F ) (1 + y) je(f )j : But fom Theoem 2.1, we know that the left-hand side is just t G (1 + x; 1 + y). Thus, changing y to y? 1, we have t G (1 + x; y) = x (F ) je(f )j y o F G x t G (1 + x; y) = F G x c(f ) y je(f )j ; (5) an equation that will be useful in the futue. Note that if G wee allowed to be disconnected then the facto of x on the left of this equation would be eplaced with x c(g). We will now give a chaacteization of the edges in E(F ) that will allow us to mine moe combinatoial infomation fom equation (5). Conside a tee T in F ooted at its smallest vetex. Then we will use all the usual family tee conventions when talking about vetices of T (paent, child, and so on). Also, we call a pai of vetices (u; v) in T an invesion (espectively non-invesion) if u is an ancesto of v and u > v (espectively u < v). Finally, a coss edge is e = fu; vg whee u is not a descendant of v and vice-vesa. Lemma 2.3 Suppose G is a gaph with spanning foest F and e 2 G? F. Then e 2 E(F ) if and only if e is of one of the following types: 1. e = fu; vg whee v is a descendant of u, and (w; v) is an invesion whee w is the child of u on the unique u-v path in F, o 2. e < f whee f 2 F is an edge with the same endpoints as e, o 3. e is a loop. Figue 2 shows a schematic diagam of an extenally active edge coesponding to an invesion. Fo a concete example, see Figue 1 whee edges f1; 2g and f3; 7g ae of type 1, edge a = f4; 6g is of type 2 and f2; 2g is of type 3. Poof of Lemma 2.3. It suces to show that F(F [ e) does not contain e if and only if e is one of these thee types. If e is of type 1, then DFS will each u befoe v. But since we ae using geatest-neighbos, the seach will continue to

8 the electonic jounal of combinatoics 3 (2) (1996), #R9 6 w u w > v A AHH v Figue 2: The invesion case of Lemma 2.3 w egadless of the pesence of fu; vg. By the time the seach eaches v, u has aleady been visited and so e cannot be used in that diection eithe. The paallel edge case follows fom using geatest edges, and loops ae neve in foests. Fo the convese, suppose that e is not one of the thee types. Then e must be of the fom i. e = fu; vg whee v is a descendant of u and (w; v) is a non-invesion, whee w is the child of u on the unique u-v path in F, o ii. e > f whee f 2 F is an edge with the same endpoints as e, o iii. e is a coss edge. In the st two cases, the geatest-neighbo seach will be foced to tavese e the st time it is encounteed. So e 2 F(F [ e). In the thid case, suppose e = fu; vg and that u is seached st in F. Then v would eventually be visited as a neighbo of u in F [ e. Again, this foces e 2 F(F [ e). 3 Oientations If G is a gaph, then an oientation O of G is a digaph obtained by assigning one of the two possible diections to each edge of G. If e = fu; vg is an edge then the coesponding ac will be denoted ~e with possible diections ~e = uv o ~e = vu. Fo enumeative puposes, we also conside each loop to have two possible diections. A suboientation of G is an oientation of a subgaph of G. A digaph is acyclic if it contains no diected cycles. Loops and oppositely diected paallel edges ae consideed cycles. We can use the Tutte polynomial and DFS to count acyclic suboientations of G. Given any digaph D, we use least-neighbo seach, which goes to the smallest vetex at each step. If thee ae paallel acs between the two vetices, they ae

9 the electonic jounal of combinatoics 3 (2) (1996), #R ??@? I 6 -? i? 6? ) ? 4 6??? 6? 7 8 D F Figue 3: A digaph D and foest F D odeed and the smallest one is taken. (In DFS2 we still always stat at the least unvisited vetex.) Of couse, we only tavese acs in the pope diection. If D has least-neighbo DFS foest F, then we wite ~ F? (D) = F. The tees geneated by the least-neighbo seach of D ae elated to cetain components of the digaph. We call a digaph initially connected if thee is a diected path fom the smallest vetex to any othe. An abitay digaph can be decomposed into initial components as follows. The st component contains all vetices eachable by a diected path fom the least vetex. Remove these vetices and the second component will contain vetices eachable fom the smallest vetex that emains, etc. The digaph in Figue 3 has thee initially connected components, namely the subdigaphs induced by the vetex sets f1; 2; 3; 5g; f4; 6; 8g and f7g. Note that all acs between two such components ae diected fom the late to the ealie component. In geneal, we say that an ac uv is diected late to ealie if u is visited afte v in DFS. We will wite c(d) fo the numbe of initial components of D. Notice that if D has DFS foest F, then c(d) = c(f ). Also, c(f ) coincides with the numbe of components of F consideed as an undiected gaph (sometimes called the weak components). Given a digaph D containing a foest F, the acs uv 2 D?F can be patitioned into fou types: Fowad acs whee u is an ancesto of v, Backwad acs whee u is a descendant of v, Coss acs whee u is neithe an ancesto no a descendant of v, and Loops.

10 the electonic jounal of combinatoics 3 (2) (1996), #R ? 3? 6 A a AA B 2 BB 8 A 5 7 b 4 1? 3 BB 6?A? N a?? AA U B 2 BB b 8 A ? 3? B B?A BB N AA U 2 B A B a? 8 G O G F = ~ F? (O) Figue 4: A gaph G, suboientation O, and least-neighbo foest F Fo example, with espect to the digaph D and subfoest F of Figue 3 we have fowad ac 12; backwad acs 31, 64; coss acs 42, 75, 78, 86; and a loop at the vetex 8. It will be convenient to keep this patition in mind in the futue. Now suppose G is a gaph and F G is a foest. Conside F to be a suboientation of G, whee each edge is diected away fom the oot of its tee. Let ~ E? (F ) be the set of all oientations ~e of edges in G? F such that 1. F [ ~e is acyclic, and 2. ~F? (F [ ~e ) = F. We call the acs in ~ E? (F ) (diected) least-neighbo extenally active. Note that by the condition 1, ~ E? (F ) neve contains any loops. Also, ~ E? (F ) contains at most one oientation of each edge of G which is not a loop: if e = fu; vg with u an ancesto o descendant of v then ~e must be oiented fowad by condition 1; if e is a coss edge then it must be diected late to ealie by condition 2. If we conside the gaph G and suboientation O in Figue 4, then the acs in ~ E? (F ) ae 15, 41, 64, 75 and b oiented in the diection 68. The analog of Poposition 2.2 in this setting is as follows. Poposition 3.1 If O is any suboientation and F is any spanning foest of G then O is acyclic and ~ F? (O) = F () F O F ] ~ E? (F ): Poof. Fist suppose that O is acyclic and that F ~? (O) = F. Then clealy F O. If O = F ] E ~? (F ), then some ~e 2 O must violate condition 2 (since O is acyclic by assumption). But this implies F ~? (O) 6= F, a contadiction. Fo the othe diection, condition 2 implies F ~? (O) = F. To veify acyclicity, suppose to the contay that v 1 ; v 2 ; : : : ; v k ; v 1 is a diected cycle in O. If all the

11 the electonic jounal of combinatoics 3 (2) (1996), #R9 9 v i ae on the same path fom the oot of some tee of F, then some cycle ac is oiented backwad, contadicting the obsevation afte the denition of ~ E? (F ). Othewise we have a coss ac, say v k v 1. We can assume that all coss acs have endpoints in the same tee of F, since once we have left a component by a coss ac we can neve etun. Note that v 1 is ealie that v k because all coss acs ae diected late to ealie. Also v 1 is not an ancesto of v k by denition of coss ac. Assume, by induction, that v i?1 is ealie than v k and not v k 's ancesto. Now v i?1 v i must be eithe a fowad o coss ac. In the fome case, v i is ealie than and not an ancesto of v k. In the latte case, v i must be ealie than v i?1 and theefoe ealie than v k. Also, v i cannot be an ancesto of v k : Since v i?1 v i is a coss ac, evey descendant of v i is ealie than v i?1, but by the induction hypothesis v k is late than v i?1. Thus the induction hypothesis holds fo v i, which is a contadiction when i = k. Next, we chaacteize the acs in ~ E? (F ) just as we did fo E(F ). The poof is simila to that of Lemma 2.3 and is left to the eade. Lemma 3.2 Suppose G is a gaph with spanning foest F and e 2 G? F. Then ~e 2 ~ E? (F ) if and only if ~e is of one of the following types: 1. ~e = uv is a fowad ac, and (w; v) is a non-invesion whee w is the child of u on the unique u-v path in F, o 2. ~e > ~ f whee ~ f 2 F is an edge with the same endpoints and oientation as ~e, o 3. ~e is a coss ac diected late to ealie. In ou pevious example the ac 15 is of type 1, ac b = 68 is of type 2, and all othe acs in ~ E? (F ) ae of type 3. Compaing Lemmas 2.3 and 3.2 (in paticula, the conditions in the poof of the convese of the fome and in the statement of the latte), we immediately obtain a coollay. Coollay 3.3 Suppose G is a gaph with spanning foest F. Then jgj = jf j + je(f )j + j ~ E? (F )j: We ae now eady to count suboientations by initial components and numbe of edges. If we did not assume that G was connected, the facto of xy n?1 on the ight side of the following equation would be eplaced by x c(g) y n?c(g).

12 the electonic jounal of combinatoics 3 (2) (1996), #R9 10 Theoem 3.4 If G has n vetices, then O x c(o) y joj = xy n?1 (1 + y) (G) t G 1 + x + x y ; y whee the sum is ove all acyclic suboientations of G. Poof. Using Poposition 3.1, Coollay 3.3, and the fact that jf j = n? c(f ) fo any spanning foest F G, x c(o) y joj = x c(f ) y jf j y jaj ~F? (O)=F A ~ E? (F ) = x c(f ) y jf j (1 + y) j~ E? (F )j = x c(f ) y jf j jgj?jf j?je(f )j (1 + y) = x c(f ) y n?c(f ) jgj?n+c(f )?je(f )j (1 + y)! c(f )! je(f )j x(1 + y) 1 = y n (1 + y) jgj?n : y 1 + y Summing ove all F, we obtain fom equation (5), O which agees with (6). x c(o) y joj = y n jgj?n x(1 + y) (1 + y) y x(1 + y) t G 1 + ; y!! y We can ewite this theoem using the same invaiants as fo subgaphs. Fo any digaph D on n vetices, let (D) = c(d)? 1 (6) and Now eplace x by xy in equation (6) o O (D) = jdj? n + c(d): x c(o) y joj+c(o) = xy n (1 + y) (G) t G 1 + xy + x; O x (O) y (O) = (1 + y) (G) t G 1 + x + xy;! y! y (7)

13 the electonic jounal of combinatoics 3 (2) (1996), #R9 11 Seveal special cases ae of inteest. To count acyclic suboientations O by edges without egad to the numbe of initial components, we set x = 1 in (6) and obtain Similaly, setting x = 1 in (7) we get In paticula, O O 2 (G) t G 3; 1 2 y joj = y n?1 (1 + y) (G) t G y ; y y (O) = (1 + y) (G) t G 2 + y; O!! 1 : 1 + y = numbe of acyclic suboientations of G. On the othe hand, counting such oientations by numbe of initial components is done by putting y = 1 in (6): x c(o) = x2 (G) t G 1 + 2x; 1 : 2 To count those O fo which c(o) = 1, i.e., those which ae initially connected, we put x = 0 in (7) and obtain! y joj 1 = y n?1 (1 + y) (G) t G 1; : 1 + y c(o)=1 In paticula, 2 (G) t G 1; 1 = numbe of initially connected acyclic suboientations of G. 2 Finally, to count acyclic oientations of G, i.e., those O with joj = jgj, we divide (6) by y jgj and take y! 1. The esult is In paticula, joj=jgj x c(o) = x t G (1 + x; 0) : (8) t G (2; 0) = numbe of acyclic oientations of G, and t G (1; 0) = numbe of initially connected acyclic oientations of G. This intepetation of t G (2; 0) was found by Stanley in [38], while Geene and Zaslavsky [21] discoveed the one fo t G (1; 0). These authos expessed thei esults in tems of chomatic polynomials. :

14 the electonic jounal of combinatoics 3 (2) (1996), #R Subdigaphs Let G be a gaph. A diected subgaph o subdigaph of G is a digaph D that contains up to one copy of each oientation of evey edge of G. Thus we pemit both oientations of an edge (including loops) to appea in a subdigaph, as opposed to a suboientation whee only one is pemitted. We now conside geatest-neighbo DFS on the set of all subdigaphs of G. The only dieence fom the subgaph case is that now we ae constained to follow the diections on the acs. If digaph D has geatest-neighbo foest F, we wite ~F + (D) = F. Thee is also the set ~ E + (F ) of (diected) geatest-neighbo extenally active oientations ~e of edges of G such that ~ F + (F ] ~e ) = F. Notice that because of the disjoint union, we have ~e 2= F. Howeve, if uv 2 F then we always have vu 2 ~ E + (F ). The next fou esults ae simila to those we have seen in the pevious sections, so we will only indicate a poof of the thid. In what follows, if F is a foest then an E-active edge is an edge e in E(F ), i.e., e is geatest-neighbo extenally active in the undiected sense. All othe edges of G? F will be called E-passive. Poposition 4.1 If D is any subdigaph and F is any spanning foest of G then ~F + (D) = F () F D F ] ~ E + (F ): Lemma 4.2 Suppose G is a gaph with spanning foest F. Then ~e 2 ~ E + (F ) if and only if ~e is of one of the following types: 1. e is an E-active ac diected fowad. 2. e is any ac of G diected late to ealie. Coollay 4.3 Suppose G is a gaph with spanning foest F. Then jgj = j ~ E + (F )j? je(f )j: Poof. The edges of G can be patitioned into those in F, those that ae E-active and those that ae E-passive. The following table lists the numbe of times each sot of edge is counted in ~ E + (F ) and E(F ). edges E ~ + (F ) E(F ) F 1 (backwad) 0 E-active 2 (fowad and backwad) 1 E-passive 1 (late to ealie) 0

15 the electonic jounal of combinatoics 3 (2) (1996), #R9 13 Since the net dieence is 1 in each case, the esult follows. Theoem 4.4 If G has n vetices, then and D x c(d) y jdj = xy n?1 (1 + y) jgj t G 1 + x y ; 1 + y! D whee the sum is ove all subdigaphs of G. x (D) y (D) = (1 + y) jgj t G (1 + x; 1 + y) (10) As special cases, we can count subdigaphs by edges o invaiant: y jdj = y n?1 (1 + y) jgj t G 1 + 1! y ; 1 + y = (1 + y) 2jGj D D y (D) = (1 + y) jgj t G (2; 1 + y) 2 jgj t G (2; 2) = numbe of subdigaphs of G = 4 jgj ; o by numbe of initial components: D c(d)=1 x c(d) = x2 jgj t G (1 + x; 2) y jdj = y n?1 (1 + y) jgj t G (1; 1 + y) 2 jgj t G (1; 2) = numbe of initially connected subdigaphs of G. Fom equations (4) and (10), we see that x (D) y (D) = (1 + y) jgj x (H) y (H) : D This equality can also be poved diectly by exhibiting a 2 jgj -to-1 map fom subdigaphs D of G to subgaphs H G that peseves the appopiate invaiants as follows: Fom Lemma 4.2, D can be epesented by a tiple (F; E; A) whee F = ~ F + (D), E is the set of diected E-active edges in D, and A is the est of the acs of D (so the coesponding edges could be an abitay subset of G). Similaly, Lemma 2.3 shows that H can be epesented by a pai (F; E) whee F = F + (H) and E is the set of E-active edges in H. It is easy to veify that the pojection map (F; E; A)! (F; E) HG (whee we change acs to edges in the image) has the desied popeties. (9)

16 the electonic jounal of combinatoics 3 (2) (1996), #R Complete gaphs We will now show how ou esults on oientations and subdigaphs can be combined with the geneating function fo the Tutte polynomial of a complete gaph to obtain vaious new geneating functions, some of which genealize esults aleady in the liteatue. Fo bevity, let t n (x; y) = t Kn (x; y). Tutte [41, equation (17)] obtained an equation equivalent to the following which can be deived using the exponential fomula. Theoem 5.1 The Tutte polynomial of the complete gaph has exponential geneating function n1 t n (x; y) un n! = 1 x? 1 82 >< 4 >: n0 y 2) (n (y? 1)?n un n! 3 5 (x?1)(y?1) Next we nd the geneating function fo acyclic digaphs, which ae just acyclic suboientations O K n. To do this, it will be convenient to dene the gaphic geneating function of a sequence (a n ) n0 to be n0 a n u n (1 + y) (n 2) n! : Accoding to equation (6), the count of acyclic digaphs on n vetices by numbe of acs and initial components is given by! a n (x; y) def = x c(o) y joj = xy n?1 (1 + y) (n?1 2 ) tn 1 + x + x OK n y ; 1 : 1 + y Applying the pevious theoem yields n1 a n (x; y) y n (1 + y) (n?1 2 ) u n n! = x y 1 x + x y = y 82 >< 4 >: n0 82 < 4 : n0 1 (1 + y) (n 2) n (1 + y)n (?1) y n?y!?n u n 1 + y n! n u (1 + y) (n 2) n! 3 5? 1 3 5?x 9 >= >; : (x+ x?y )( y 9 = 1+y )? 1 9 >= >;?1 ; : (11) If we dene a 0 (x; y) = 1 and eplace u by yu, then this last esult simplies. 1+y Coollay 5.2 The gaphic geneating function fo acyclic digaphs is n0 u n a n (x; y) = (1 + y) 2) (n n! 2 4 (?1) n u n n0 (1 + y) (n 2) n! 3 5?x :

17 the electonic jounal of combinatoics 3 (2) (1996), #R9 15 Stanley [38] and Robinson [35] obtained this esult when x = y = 1, as did Liskovets [29] and Rodionov [36] when x = 1. When x and y ae integes with y 0, Stanley also gives an intepetation to these gaphic geneating functions in tems of the theoy of posets of full binomial type developed by himself, Doubilet and Rota [11]. We can deive the geneating function fo initially connected acyclic digaphs counted by numbe of acs using c n (y) def = a n(x; y) : x x=0 Dividing equation (11) by x and letting x! 0 yields the desied fomula. Coollay 5.3 The gaphic geneating function fo initially connected acyclic digaphs is n1 u n c n (y) = ln (1 + y) 2) (n n! 2 4 (?1) n u n n0 (1 + y) (n 2) n! Putting togethe these last two theoems, we see that n0 u n a n (x; y) = exp (1 + y) 2) (n n! 2 4 x n1 u n 3 5?1 c n (y) (1 + y) 2) (n n! This is a special case of a moe geneal exponential fomula fo digaphs which does not seem to have been stated befoe. Theoem 5.4 (Exponential fomula fo digaphs) Let D be a class of initially connected labeled digaphs with the popety that an ode-peseving change of labels does not aect membeship in D. Let c n (y) count digaphs in D on the label set f1; 2; : : : ; ng by numbe of acs. Then exp 2 4 x n1 u n c n (y) (1 + y) 2) (n n! 3 5 = k;m;n0 : u n 3 5 b k;m;n x k y m (12) (1 + y) 2) (n n! whee b k;m;n is the numbe of digaphs on f1; 2; : : : ; ng with m acs, k initial components, and evey such component in D. Poof. Taking the coecient of x k u n on both sides of equation (12), it suces to show 1 b k;m;n y m = 1 ky c ni (y) (1 + y) 2) (n n! k! m0 n 1 ++n k =n i=1 (1 + y) (n i 2 ) ni!

18 the electonic jounal of combinatoics 3 (2) (1996), #R9 16 whee the sum is ove all odeed patitions of n. Equivalently b k;m;n y m = 1! n c n1 (y) c nk (y) (1 + y) k! m0 n 1 ++n k =n n 1 ; : : : ; n k P i<j n in j The left side of this fomula just counts digaphs D on n vetices with k initial components by numbe of acs. But the ight sums to the same thing. The multinomial coecient counts the numbe of ways to patition the vetices of D into k odeed subsets fo the initial components. Summing ove all odeed patitions of n and then dividing by k! gives coecients which count unodeed patitions of the vetices. The c ni (y) give the ac count fo each component. And the powe of 1 + y accounts fo the acs between components which must all be diected fom a late to an ealie component. A geneal theoy of exponential fomulas has been developed by Stanley [39], but this esult does not seem to be a consequence. In the example we have been consideing, the set D consists of all initially connected acyclic digaphs. As a futhe demonstation, we can count digaphs without the acyclicity condition. Let d n (x; y) = P D xc(d) y jdj (espectively, e n (y) = P D yjdj ) whee the sum is ove all digaphs (espectively, all initially connected digaphs) on n vetices. The gaphic geneating function fo all digaphs by numbe of acs is n0 u (1 + y) 2(n 2) n = (1 + y) 2) (1 + y) 2) (n un (n n! n! n0 Using ou Exponential Fomula, we immediately get the following esult. Coollay 5.5 The gaphic geneating function fo d n (x; y) is n0 u n d n (x; y) = (1 + y) 2) (n n! The gaphic geneating function fo e n (y) is n1 u n e n (y) = ln (1 + y) 2) (n n! 6 Neighbos-st seach 2 4 n0 2 4 n0 (1 + y) 2) (n un n! x (1 + y) 2) (n un 5 : n! A specialization of t n (x; y) that has eceived some attention is t n (y) def = t n (1; y). Mallows and Riodan [30] st studied this polynomial as the invesion enumeato fo tees. See also the book of Foata [12, pp. 144{147] and the papes of :

19 the electonic jounal of combinatoics 3 (2) (1996), #R9 17 Gessel, Sagan and Yeh [19] and Gessel [17]. Keweas [28] has given a numbe of othe intepetations to this polynomial which have been futhe studied by Moszkowski [31]. See also Beissinge [3], who gives a bijective poof that t n (1; y) counts invesions. In Section 9 we will give a table of these polynomials. Fo completeness, we state the connection with the invesion polynomial. It follows diectly fom Lemma 2.3 applied to G = K n. Poposition 6.1 If T is a tee, let inv T stand fo the numbe of invesions of T. Then t n (y) = y inv T T whee the sum is ove all tees with n vetices. We will now give anothe intepetation of t n (y) using a modied vesion of DFS which is sometimes called neighbos-st seach o NFS (see [7, p. 154]). The following steps ae applied to a gaph H to build an NFS foest F. Note that making and seaching a vetex ae now two sepaate actions. NFS1 Let F = ;. NFS2 Let v be the least unmaked vetex in V and mak v. NFS3 Seach v by making all neighbos of v that have not been maked and adding to F all edges fom v to theses vetices. NFS4 Recusively seach all the vetices maked in NSF3 in inceasing ode, stopping when evey vetex that has been maked has also been seached. NSF5 If thee ae unmaked vetices, then etun to NSF2. Othewise, stop. Thus NFS seaches nodes in a depth-st manne but maks childen in a locally beadth-st manne. In choosing the vetex u in NFS2, we will always pick the one with smallest label and use the smallest odeed edge. Denote the esulting foest by F = F N (H). As an example, fo the gaph H in Figue 5 we stat at vetex 1, designating 3 and 4 as its childen. Next we seach node 3 and mak 5, 6 and 8 as its osping. Note that 4 cannot be a child of 3 since it is aleady a child of 1. The seach now continues at 5, and so foth. Obseve that tavesing a foest F by NFS gives a linea odeing to the childen of each vetex, i.e., the ode in which we seach them fom smallest to lagest label. We will display this as a left-to-ight ode of the siblings when we daw F in the plane and use coesponding teminology.

20 the electonic jounal of combinatoics 3 (2) (1996), #R9 18 3?@ 4? H ?@? F = F N (H) Figue 5: A gaph H and NSF foest F As usual, given a spanning foest F of a gaph G, we dene E N (F ), the set of edges extenally active with espect to NFS, to be those edges e in G? F such that F N (F [ e) = F: The next set of esults should be easy fo the eade to pove by mimicking what we did in Section 2. Poofs ae theefoe omitted. Poposition 6.2 If H is any subgaph and F is any spanning foest of G then F N (H) = F () F H F ] E N (F ): Poposition 6.3 If G is a connected gaph, then t G (1 + x; y) = x (F ) y je N(F )j F G whee the sum is ove all spanning foests of G. In paticula t n (y) = T y je N (T )j (13) whee the sum is ove all tees on n vetices. Theoem 6.4 Suppose G is a gaph with spanning foest F and e 2 G? F. Then e 2 E N (F ) if and only if e is of one of the following types: 1. e = fu; vg whee v is a descendant of u's paent, and w < u whee w is the sibling of u on the unique path fom thei paent to v in F, o

21 the electonic jounal of combinatoics 3 (2) (1996), #R9 19 w H H u w < u A AHH v Figue 6: The st case of Theoem e > f whee f 2 F is an edge with the same endpoints as e, o 3. e is a loop. Since ou applications will all be to G = K n, only the st of these thee cases eally mattes. A schematic diagam of this case is given in Figue 6. It follows fom Popositions 6.1 and 6.3 that the distibution of je N (T )j fo labeled NFT tees is the same as that fo inv T. We digess biey to note that the the distibution of extenal activities fo unlabeled odeed tees is given by the q-catalan numbes studied by Andews [1], Fulinge and Hofbaue [14], and Kattenthale [27]. Any unlabeled odeed tee can be given an NFT labeling by labeling the oot as 1 and then making sue that the labels on the childen of evey vetex incease fom left to ight. Thus we can let the extenal activity of an unlabeled tee T be the extenal activity of any NFT labeling of T as a spanning tee of a complete gaph. Now dene polynomials C n (q) by C 0 (q) = 1 and The st few values ae C n (q) = n?1 k=0 q k C k (q)c n?k?1 (q): (14) C 0 (q) = C 1 (q) = 1; C 2 (q) = 1 + q; C 3 (q) = 1 + 2q + q 2 + q 3 ; C 4 (q) = 1 + 3q + 3q 2 + 3q 3 + 2q 4 + q 5 + q 6 : If we compute the extenal activities of the unlabeled odeed tees on 3 edges (see Figue 7), then we obtain jt j=3 This is evidence fo the next theoem. q je N(T )j = 1 + 2q + q 2 + q 3 :

22 the electonic jounal of combinatoics 3 (2) (1996), #R9 20 @@???? je N (T )j : Figue 7: Extenally active edges fo unlabeled odeed tees on Theoem 6.5 We have C n (q) = jt j=n q je N (T )j whee the sum is ove all unlabeled odeed tees T with n edges. Poof. It suces to show that the tee sum satises the ecusion (14). Now any tee T can be decomposed into two tees T 0 and T 00 whee T 0 = ightmost subtee of the oot, and T 00 = T? T 0 Hee we make the convention that the edge joining to T 0 is emoved in T? T 0. Also, if has only one subtee, it is consideed ightmost so that T 00 = in this case. But if jt j = n and jt 00 j = k, then jt 0 j = n? k? 1. So by case 1 of Theoem 6.4 q je N (T )j = q k q je N (T 0 )j q je N (T 00 )j : Summing ove all T, we obtain the desied esult by induction. We now etun to the main steam of ou development. Fo the est of this section and the next, all of ou extenal activities will be with espect to the gaph K n. In this case, given any tee T, we can deive a simple fomula fo je N (T )j. Let v 1 ; v 2 ; : : : ; v n be the ode in which the nodes of T ae st seached using NFS. Note that this is a depth-st ode. Dene the pex code of T to be the sequence c(t ) = c 1 ; c 2 ; : : : ; c n whee c i is the numbe of childen of vetex v i. We could also dene c(t ) ecusively by c(t ) = c 1 ; c(t 1 ); c(t 2 ); : : : ; c(t k )

23 the electonic jounal of combinatoics 3 (2) (1996), #R9 21 whee T 1 ; T 2 ; : : : ; T k ae the subtees of v 1 listed in ode of inceasing labels of thei oots. Fo example, the nodes of the tee in Figue 5 ae seached in the ode v 1 ; v 2 ; : : : ; v 8 = 1; 3; 5; 7; 6; 8; 4; 2 (15) which gives it pex code c 1 ; c 2 ; : : : ; c 8 = 2; 3; 1; 0; 0; 0; 1; 0: Theoem 6.6 If T is a tee with pex code c 1 ; c 2 ; : : : ; c n then je N (T )j = (c 1? 1) + (c 1 + c 2? 2) + + (c 1 + c c n?1? n + 1) = (n? 1)c 1 + (n? 2)c c n?1? n 2! Using ou pevious example n?1 k=1(n? k)c k? n 2 while! = ? 8 2 = 10 E N (T ) = ff4; 3g f4; 5g f4; 6g f4; 7g f4; 8g f6; 5g f6; 7g f8; 5g f8; 6g f8; 7gg which has 10 elements. Poof of Theoem 6.6. It suces to show that the tem c 1 + c c i? i counts all extenally active edges whose left end is v i+1. We will do this by induction on i. This is clealy tue fo i = 0. Fo i > 0, we distinguish two cases. If c i > 0, then v i+1 is the leftmost child of v i. Now fv i ; ug active implies that so is fv i+1 ; ug (Theoem 6.4), yielding c 1 + c c i?1? i + 1 edges. Also, thee ae active edges fom v i+1 to each of its siblings, fo c i? 1 moe edges. These ae the only active edges and the total is coect. If c i = 0, then v i is a leaf and we get to v i+1 by backtacking. But then v i+1 was the leftmost of all the vetices joined to v i by extenally active edges. So v i+1 has exactly one less (= c i? 1) active edge than v i did. Thus we ae nished by induction. We can use the NFS intepetation of t n (y) to give a combinatoial poof of an identity fo its geneating function st poved by othe means in [16].!

24 the electonic jounal of combinatoics 3 (2) (1996), #R9 22 Theoem 6.7 Let then J(u) = n0 J(u) = n0 t n+1 (y) un n! y (n 2) J(u)J(yu) J(y n?1 u) un n! Poof. Taking the coecient of un n! on both sides, we get the equivalent statement t n+1 (y) = = k0 n 1 ++n k +k=n k0 n 1 ++n k +k=n y (k 2) h y (k?1)n 1 t n1 +1(y) i h y (k?2)n 2 t n2 +1(y) i n! n 1!n 2! n k!k!! n k t n1 +1(y)t n2 +1(y) yp (k?i)(n i=1 i+1) : k; n 1 ; n 2 ; : : : ; n k whee the facto involving t ni +1(y) comes fom J(y k?i u). To see that this last expession enumeates tees T by extenally active edges, conside the subtees T 1 ; T 2 ; : : : ; T k of the oot of T. Suppose these tees have oots w 1 ; w 2 ; : : : ; w k and n 1 ; n 2 ; : : : ; n k othe vetices espectively. Then the multinomial coecient counts the numbe of ways to pick the oots and then the othe sets of vetices. The active edges fv; wg ae of two types: edges whee v and w ae in the same T i, and edges whee v 2 T i and w = w j fo some i < j Edges of the st sot ae accounted fo by t n1 +1(y)t n2 +1(y) t nk +1(y) while those of the second ae taken cae of by the powe of y. We end with a chaacteization of foests in tems of pex codes that will help us in Section 7. Since it is well known we omit the poof. Theoem 6.8 The sequence c 1 ; c 2 ; : : : ; c n is a pex code fo a tee if and only if j i=1 n i=1 (c i? 1) 0 fo j < n, and (c i? 1) =?1

25 the electonic jounal of combinatoics 3 (2) (1996), #R9 23 Notice that the peceding conditions could be ewitten as j n i=1 i=1 c i j fo j < n, and c i = n? 1: We can make the chaacteization in Theoem 6.8 even stonge by using paent functions. Suppose we ae given a tee T having vetices f1; : : : ; ng and NFS ode v 1 = 1; v 2 ; : : : ; v n. The coesponding paent function is p : f2; : : : ; ng! f1; : : : ; ng dened by p(i) = j if the vetex labeled i has paent v j. Retuning to the tee in Figue 5 with NFS ode given by (15), we see that it has paent function p(2) = 7; p(3) = 1; p(4) = 1; p(5) = 2; p(6) = 2; p(7) = 3; p(8) = 2: Obseve that jp?1 (j)j = c j. This should motivate the following esult whose poof, since it follows easily fom the pevious theoem, is omitted. Coollay 6.9 The function p : f2; : : : ; ng! f1; : : : ; ng is a paent function fo a tee if and only if jp?1 (1) [ p?1 (2) [ [ p?1 (j)j j fo j < n 7 Hashing and paking functions We now descibe an application of t n (y) to hash coding, which is a method fo stoing and etieving data eciently. Knuth [24, Chapte 6] gives a compehensive account of stoage and etieval methods, and in paticula of hash coding (Section 6.4). The hashing technique we discuss hee is called \open addessing with linea pobing." It was analyzed ealie by Knuth in [25]. We conside only the stoage aspect; etieval is simila and is discussed in detail by Knuth. Suppose we have m boxes labeled 1 to m, and n < m objects labeled 1 to n which ae to be put into the boxes. Each object i has a pefeed box h(i), whee the function h is called a hash function. We now inset the objects in the ode 1, 2,..., n into the boxes. When we inset object i, we place it into box h(i) if this box is empty. Othewise, we pobe boxes h(i) + 1; h(i) + 2; : : : in tun and place object i into the st empty box we nd. Box numbes ae taken modulo m, so that box 1 is pobed afte box m. Since thee ae moe boxes than objects, evey object will eventually be placed into a box. Given a hash function h, which may be an abitay function fom f1; 2; : : : ; ng to f1; 2; : : : ; mg, we let B(h) be the

26 the electonic jounal of combinatoics 3 (2) (1996), #R Figue 8: Open addess hashing numbe of times an occupied box is pobed duing the insetion pocess. (Note that Knuth counts as a pobe the box into which an object is inseted, but we do not.) By way of illustation, conside the aay of m = 9 boxes in Figue 8 whee the box numbes ae given on the fa left. Reading the diagam fom left to ight shows the placement of n = 6 objects using the hash function h(1) = 9; h(2) = 3; h(3) = 3; h(4) = 9; h(5) = 4; h(6) = 3: The numbe of pobes is B(h) = = 6: The following lemma, which appeas in [24, pp. 530{531], is vey useful. Lemma 7.1 (Reaangement Lemma) Suppose h and g ae two hash functions such that the sequence g(1); : : : ; g(n) is a eaangement of h(1); : : : ; h(n). Then 1. the same boxes ae lled in the insetion pocess fo h and g, and 2. B(h) = B(g). We now study the distibution of B(h) among the m n possible hash functions h : f1; 2; : : : ; ng! f1; 2; : : : ; mg, whee n < m. Let K n;m;i be the numbe of such h with B(h) = i. Also, let L n;m;i be the numbe of these functions with the popety that afte all n objects ae inseted, box m is empty. Since all boxes ae equally likely to be empty, we have L n;m;i = k m K n;m;i; (16)

27 the electonic jounal of combinatoics 3 (2) (1996), #R9 25 whee k = m? n is the numbe of empty boxes. We now examine the polynomial L n;m (y) = i0 L n;m;i y i : (17) Suppose we pefom the insetions coesponding to a hash function counted by L n;m (y). Conside the sequence of boxes afte completing these insetions. This sequence can be boken up into k = m? n subsequences, each of which consists of zeo o moe lled boxes followed by an empty box. Since no object will eve pobe any of the k empty boxes, the sequence of boxes can be obtained by decomposing the hash function into k functions and doing the insetions fo each sepaately. In the example fom Figue 8, thee ae k = 3 subsequences, consisting of boxes f3; 4; 5; 6; 7g, f8g, and f9; 1; 2g. By the Reaangement Lemma and the popeties of exponential geneating functions, we have n0 L n;n+k (y) un n! = 2 4 n0 L n;n+1 (y) un n! 3 5 k : (18) It emains to detemine L n;n+1 (y). The functions counted by L n;n+1 (y) ae called paking functions: they ae hash functions p : f1; : : : ; ng! f1; : : : ; n + 1g that leave box n + 1 empty. The name deives fom the scenaio [24, p. 545, execise 29] in which the boxes ae intepeted as paking spaces and the objects ae cas tying to pak, with the hash function giving the pefeed spot of each ca. The tem \paking function" was coined by Konheim and Weiss [26]. We have the following chaacteization of paking functions. Theoem 7.2 The hash function p : f1; : : : ; ng! f1; : : : ; n + 1g is a paking function if and only if Futhemoe, in this case whee c i = jp?1 (i)j. jp?1 (1) [ p?1 (2) [ [ p?1 (j)j j fo j < n + 1: B(p) = nc 1 + (n? 1)c c n? n Poof. Suppose that p is a paking function. Since the st j boxes can be lled only fom objects in p?1 (1) [ p?1 (2) [ [ p?1 (j), we must have jp?1 (1) [ p?1 (2) [! ;

28 the electonic jounal of combinatoics 3 (2) (1996), #R9 26 [ p?1 (j)j j fo j < n + 1. The convese follows fom the obsevation that if jp?1 (1) [ p?1 (2) [ [ p?1 (j)j j then box j will be lled. To pove the fomula fo B(p) it suces to show that c 1 + c 2 + : : : + c j? j counts the numbe of times box j is pobed afte it is lled, fo j = 1; 2; : : : ; n. But c 1 + c c j is the total numbe of objects that stat thei seach in box j o befoe. And of these, the st j objects will occupy the st j boxes, leaving c 1 + c 2 + : : : + c j? j to pobe box j. Compaison of Theoem 7.2 with Theoem 6.6 and Coollay 6.9 shows that thee is a bijection between NFS tees T on f1; 2; : : : ; n + 1g and paking functions p : f1; : : : ; ng! f1; : : : ; n + 1g such that E N (T ) = B(p). Thus L n;n+1 (y) = t n+1 (y): (19) This was st poved by Keweas [28], who studied the functions satisfying the popety of Theoem 7.2, but did not identify them as paking functions. Fo futhe wok on paking functions, see Schutzenbege [37], Riodan [34], Foata and Riodan [13], and Moszkowski [31]. In analyzing the pefomance of hash coding as a stoage method, one wants to know the expected value of B(h) ove all hash functions h : f1; 2; : : : ; ng! f1; 2; : : : ; mg, assuming that all ae equally likely. Although Knuth computes this expected value without knowing L n;m (y), it is inteesting to see how this value can be deived fom ou esults. The expected value of B(h) ove all hash functions is clealy the same as the expected value of B(h) ove hash functions that leave box m empty, which is L 0 n;m(1)=l n;m (1). By (16) and (17), L n;n+k (1) = Also, by (18) and (19) we have n0 k L 0 n;n+k (1)un n! = k 2 4 n0 n + k (n + k)n = k(n + k) n?1 : t n+1 (1) un n! 3 5 k?1 n0 t 0 n+1 (1)un n! : (20) We know that t n+1 (1) = (n+1) n?1. It emains to evaluate t 0 n+1(1). By equation (4), we have t n+1 (y) = (y? 1) (H) H whee the sum is ove all connected gaphs on f1; 2; : : : ; n + 1g. Then t 0 n+1(1) = H (H) (y? 1) (H)?1 : y=1

29 the electonic jounal of combinatoics 3 (2) (1996), #R9 27 The only non-zeo tems in this sum occu when 1 = (H) = jhj?(n+1)+c(h). Since H is connected, this implies it must be unicyclic. The numbe of such gaphs is known [32, 43]. Substituting this value into the pevious equation gives t 0 n+1(1) = 1 2 n+1 j=3 n + 1 j! j! (n + 1) n?j : (21) Now let It is well known that T = T (u) = n0 T j 1? ut = (l + j) l ul l! : l0 n?1 un (n + 1) n! : (22) See, fo example, Riodan's book [33, p. 147]. It follows that 1 n + 1 n?j un!j! (n + 1) j n! = uj?1 T j 1? ut : n=j?1 Combining this equation with (20), (21), and (22) yields n0 L 0 n;n+k(1) un n! = k 2 T k?1 1 = k 2 = k 2 = n0 1 i=2 j=3 u i T i+k 1? ut 1 i=2 l0 u n n! u j?1 T j 1? ut (l + i + k) l ul+i l! n i=2 1 2 n i! i! k(n + k) n?i : Dividing by L n;n+k (1) = k(n+k) n?1 and setting m = n+k, we obtain the expected value. Poposition 7.3 The expected value of B(h) as h vaies ove all hash functions fom f1; 2; : : : ; ng to f1; 2; : : : ; mg (n < m) is! " n n n(n? 1) n(n? 1)(n? 2) 1 2 i=2 i i! m 1?i = 1 2 m + m 2 # + : (23)

30 the electonic jounal of combinatoics 3 (2) (1996), #R9 28 To elate Poposition 7.3 to Knuth's esults, we note that he consides the quantity C 0 n?1 which is the expected numbe of pobes to inset the nth object fo a andom hash function fom f1; 2; : : : ng to f1; 2; : : : mg. Since Knuth counts the pobe of a vacant box, which we do not, (23) is equal to P n j=1(c 0 j?1? 1). P Conveniently, he is also inteested in the quantity C n = 1 n n j=1 C0 j?1. Since (23) is equal to n(c n? 1), it is easy to check that Knuth's fomula (40) in [24, p. 530] agees with Poposition Comments and open questions Seveal aeas elated to what we have pesented deseve futhe investigation. (1) Thee ae many othe specializations of the Tutte polynomial that enumeate vaious classes of objects. See Bylawski's suvey aticle [5] o his aticle with Oxley [6] fo a list in the context of matoids. How many of these can be explained by eithe DFS? (2) Stanley's intepetation of t G (2; 0) was actually pat of a moe geneal esult [38]. He poved that if G is a connected gaph and k is a positive intege, then k t G (1 + k; 0) (24) is the numbe of pais (O; f) whee O is an acyclic oientation of G, and f : V! f1; 2; : : : ; kg is a function such that uv 2 O. implies f(u) f(v). Also we know, fom equation (8), that (24) counts pais (O; g) whee O is an acyclic oientation of G, and g : V! f1; 2; : : : ; kg is a function such that u; v in the same initial component of O implies g(u) = g(v). Recently Sege Elnitsky [pivate communication] has found a diect bijection between such pais. (3) In Theoem 6.7, we poved an identity fo J(u) = P n0 t n+1 (y)u n =n!. This is a special case of the fact [16] that J(u)J(yu) J(y k u) = (1 + y + + y k ) n y 2) (n J(u)J(yu) J(y n?1 u) un n! : n0

31 the electonic jounal of combinatoics 3 (2) (1996), #R9 29 Unfotunately, we have not been able to nd a combinatoial poof of this fomula based on counting extenally active edges. (4) Paking functions have been eceiving a lot of attention ecently because of thei connection with a poblem in epesentation theoy. The Reaangement Lemma shows that thee is an action of pemutations in the symmetic goup S n on paking functions p : f1; : : : ; ng! f1; : : : ; n + 1g given by p(i) = p(?1 i): Thus the set of paking functions can be made into an S n -module which we denote by P n. Now conside the polynomial ing R n = C[x 1 ; : : : x n ; y 1 ; : : : ; y n ] whee C is the complex numbes. Let 2 S n act on q 2 R n diagonally, i.e., q(x 1 ; : : : x n ; y 1 ; : : : ; y n ) = q(x 1 ; : : : x n ; y 1 ; : : : ; y n ): If J R n is the ideal of nonconstant invaiants of this action, then the quotient R n =J is anothe S n -module. Mak Haiman conjectued that thee is an isomophism P n = Q (Rn =J) (25) whee Q is a module fo the sign epesentation. Moeove, since the bidegee of a polynomial in the x's and y's is peseved unde the action of S n, R n =J is a bigaded S n -module. If we ignoe the y-gading then (25) seems to be an isomophism of x-gaded S n -modules, whee the degee of a paking function p is B(p) as dened in Section 7. Thee is a sizable amount of numeical evidence fo this conjectue. Howeve, it is still mysteious that two such dieently dened objects should tun out to be isomophic. Fo moe infomation about this question, see [22]. 9 Tables We will now give tables fo vaious quantities that we have studied. Tables 1a and 1b contain the Tutte polynomials of the complete gaphs t n (x; y) fo n 8. Fo n 3, the polynomials ae witten in the usual fomat. Fo 4 n 8, we let t n (x; y) = t n;i;j x i y j i;j and then display the coecients in a ectangula matix with the enty in ow j and column i of the nth aay being being t n;i;j. Table 2 gives the specializations t n (y) = t n (1; y) which ae also invesion enumeatos fo tees.

32 the electonic jounal of combinatoics 3 (2) (1996), #R9 30 t 1 (x; y) = 1 t 2 (x; y) = x t 3 (x; y) = x + x 2 + y t 4 (x; y) : t 5 (x; y) : t 6 (x; y) : jni jni jni Table 1a: Tutte polynomials of complete gaphs fo n 6