Goncarov-Type Polynomials and Applications in Combinatorics

Transcription

1 Goncarov-Type Polynomals and Applcatons n Combnatorcs Joseph P.S. Kung 1, Xnyu Sun 2, and Catherne Yan 3,4 1 Department of Mathematcs, Unversty of North Texas, Denton, TX ,3 Department of Mathematcs Texas A&M Unversty, College Staton, TX Center for Combnatorcs, LPMC Nanka Unversty, Tanjn , P.R. Chna 1 kung@unt.edu, 2 xsun@math.tamu.edu, 3 cyan@math.tamu.edu February, 2006 Key words and phrases. Goncarov Polynomal, parkng functons, lattce path, q-analog Mathematcs Subject Classfcaton. Abstract In ths paper we extend the work of [1] to study combnatoral problems va the theory of borthogonal polynomals. In partcular, we gve a unfed algebrac approach to several combnatoral objects, ncludng order statstcs of a real sequence, parkng functons, lattce paths, and area-enumerators of lattce paths, by descrbng the propertes of the sequence of Goncarov polynomals and ts varous generalzatons. 4 The thrd author was supported n part by NSF grant #DMS

2 1 Introducton The man content of ths paper s to use theory of sequences of polynomals borthogonal to a sequence of lnear operators to study combnatoral problems. In partcular, we descrbed the algebrac propertes of the sequence of Goncarov polynomals and ts varous generalzatons, whch gve a unfed algebrac approach to several combnatoral objects, ncludng (1) The cumulatve dstrbuton functons of the random vectors of order statstcs of n ndependent random varables wth unform dstrbuton on an nterval; (2) general parkng functons, that s, sequences (x 1, x 2,..., x n ) of ntegers whose order statstcs are bounded between two gven non-decreasng sequences; (3) Lattce paths that avod certan general boundares; and (4) The area-enumerator of lattce paths avodng certan general boundares. The object (2) can be vewed as a dscrete analog of (1). In lterature, objects (1) and (3) have been extensvely studed by probablstc argument and countng technques. General parkng functons wth one boundary has been studed n a prevous paper by the frst and the thrd author [2]. The contrbuton of the current paper s to put all four problems n the same umbrella, and present a unfed treatment. For object (1) and (2), the correspondng polynomal sequences s Goncarov polynomals, whch are outlned n Secton 2. Ths secton also contans an ntroducton to the theory of sequences of borthogonal polynomals. In Secton 3 we descrbe the sequences of dfference Goncarov polynomals. The combnatoral nterpretaton of dfference Goncarov polynomals s lattce paths wth one-sded boundary, whch s gven n Secton 4. Secton 5 and 6 are on q-analog of dfference Goncarov polynomals, and ts applcaton n enumeratng area of lattces paths wth one-sded boundary. The two-sded boundares for both parkng functons and lattce paths are treated n Secton 7. 2 Sequences of borthogonal polynomals and Goncarov polynomals We begn by gvng an outlne of the theory of sequences of polynomals borthogonal to a sequence of lnear functonals. The detals can be found n [1]. Let P be vector space of all polynomals n the varable x over a feld F of characterstc zero. Let D : P P be the dfferentaton operator. For a scalar a n the feld F, let ε(a) : P F, p(x) p(a) be the lnear functonal whch evaluates p(x) at a. 2

3 Let ϕ s (D), s = 0, 1, 2,... be a sequence of lnear operators on P of the form ϕ s (D) = D s b sr D r, (2.1) r=0 where the coeffcents b s0 are assumed to be non-zero. There exsts a unque sequence p n (x), n = 0, 1, 2,... of polynomals such that p n (x) has degree n and where δ sn s the Kronecker delta. ε(0)ϕ s (D)p n (x) = δ sn, (2.2) The polynomal sequence p n (x) s sad to be borthogonal to the sequence ϕ s (D) of operators, or, the sequence ε(0)ϕ s (D) of lnear functonals. Usng Cramer s rule to solve the lnear system and Laplace s expanson to group the results, we can express p n (x) by the the followng determnantal formula: p n (x) = b 00 b 10 b n0 b 00 b 01 b b 0,n 1 b 0n 0 b 10 b b 1,n 2 b 1,n b b 2,n 3 b 2,n b n 1,0 b n 1,1 1 x x 2 /2!... x n 1 /(n 1)! x n /. (2.3) Snce {p n (x)} n=0 forms a bass of P, any polynomal can be unquely expressed as a lnear combnaton of p n (x) s. Explctly, we have the expanson formula: If p(x) s a polynomal of degree n, then p(x) = d p (x), (2.4)! where d = ε(0)ϕ (D)p(x). In partcular, x n = b,n p (x), (2.5)! whch gves a lnear recurson for p n (x). Equvalently, one can wrte (2.5) n terms of formal power seres equatons, and obtan the Appell relaton e xt = n=0 p n (x)ϕ n (t), (2.6) 3

4 where ϕ n (t) = t s r=0 b srt r. A specal example of sequences of borthogonal polynomals s the Goncarov polynomals. Let (a 0, a 1, a 2,...) be a sequence of numbers or varables called nodes. The sequence of Gon carov polynomals g n (x; a 0, a 1,..., a n 1 ), n = 0, 1, 2,... s the sequence of polynomals borthogonal to the operators ϕ S (D) = D s a r sd r. r! As ndcated by the notaton, g n (x; a 0, a 1,..., a n 1 ) depends only on the nodes a 0, a 1,..., a n 1. Indeed, from equaton (2.3), we have the determnantal formula, a 1 a 2 0 a 3 0 a 0 2! 3!... n 1 0 (n 1)! a 0 1 a 2 1 a 1 2!... n 2 1 (n 2)! a g n (x; a 0, a 1,..., a n 1 ) = a 2... n 3 2 (n 3)! a n 1 x 1 x 2 x 3 x 2! 3!... n 1 x n (n 1)! r=0 From equatons (2.5) and (2.6), we have the lnear recurson and the Appell relaton x n = e xt = a n g (x; a 0, a 1,..., a 1 ) g n (x; a 0, a 1,..., a n 1 ) tn e ant. n=0 a n 0 a n 1 1 (n 1)! a n 2 2 (n 2)! Fnally, from equaton (2.4), we have the expanson formula. If p(x) s a polynomal of degree n, then ε(a )D p(x) p(x) = g (x; a 0, a 1,..., a 1 ).! The sequence of Goncarov polynomals possesses a set of specfc propertes, whch are lsted n the followng. The proofs can be found n [1].. 4

5 1. Dfferental relatons. The Gon carov polynomals can be equvalently defned by the dfferental relatons Dg n (x; a 0, a 1,..., a n 1 ) = ng n 1 (x; a 1, a 2,..., a n 1 ), wth ntal condtons g n (a 0 ; a 0, a 1,..., a n 1 ) = δ 0n. 2. Integral relatons. g n (x; a 0, a 1,..., a n 1 ) = n = 3. Shft nvarant formula. x a 0 g n 1 (t; a 1, a 2,..., a n 1 )dt x t1 tn 1 dt 1 dt 2 dt n. a 0 a 1 a n 1 g n (x + ξ; a 0 + ξ, a 1 + ξ,..., a n 1 + ξ) = g n (x; a 0, a 1,..., a n 1 ). 4. Perturbaton formula. g n (x; a 0,..., a m 1, a m + b m, a m+1,..., a n 1 ) = g n (x; a 0,... a m 1, a m, a m+1,..., a n 1 ) g n m (a m + b m ; a m, a m+1,..., a n 1 )g m (x; a 0, a 1,..., a m 1 ). m Applyng the perturbaton formula repeatedly, we can perturb any subset of nodes. For example, the followng formula allows us to perturb an ntal segment of length n m + 1 : g n (x; a 0 + b 0, a 1 + b 1,..., a n m + b n m, a n m+1,..., a n 1 ) = g n (x; a 0, a 1,..., a n m, a n m+1,..., a n 1 ) n m g n (a + b ; a, a +1,..., a n 1 )g (x; a 0 + b 0, a 1 + b 1,..., a 1 + b 1 ). 5. Bnomal expanson. g n (x + y; a 0,..., a n 1 ) = g n (y; a,..., a n 1 )x. 5

6 In partcular, g n (x; a 0,..., a n 1 ) = g n (0, a,..., a n 1 )x. That s, coeffcents of Goncarov polynomals are constant terms of (shfted) Goncarov polynomals. 6. Combnatoral representaton. Let u be a sequence of non-decreasng postve ntegers. A u- parkng functon of length n s a sequence (x 1, x 2,..., x n ) whose order statstcs (the sequence (x (1), x (2),..., x (n) ) obtaned by rearrangng the orgnal sequence n non-decreasng order) satsfy x () u. Goncarov polynomals form a natural bass of polynomals for workng wth u-parkng functons. Explctly, we have P n (u 1, u 2,..., u n ) = g n (x; x u 1, x u 2,..., x u n ) = g n (0; u 1, u 2,..., u n ) = ( 1) n g n (0; u 1, u 2,..., u n ). For more propertes and computatons of parkng functons va Goncarov polynomals, please refer to [1, 2, 3]. In partcular, the sum enumerator and factoral moments of the sums are computed. For u-parkng functons, the sum enumerator s a specalzaton of g n (x; a 0, a 1,..., a n 1 ) wth a = 1 + q + + q u 1. Generatng functons for factoral moments of sums of u-parkng functons are gven n [1], whle the explct formulas for the frst and second factoral moments of sums of u-parkng functons are gven n [2], and n [3] for all factoral moments for classcal parkng functons where u forms an arthmetc progresson. Remark. Sequences of polynomals of bnomal type and the related Sheffer sequences can be vewed as specal cases of sequences of borthogonal polynomals. We shall use a descrpton gven n the classcal paper of Rota, Kahaner and Odlyzko [9]. A delta operator B s a formal power seres of order 1 n the dervatve operator D, B(D) = D + b 2 D 2 + b 3 D 3 + A Sheffer sequence {s n } (for B) s a polynomal sequence such that Bs n = s n 1 for all n = 0, 1, 2,..., The basc sequence {b n } (for B) s the Sheffer sequence wth ntal values b n (0) = δ 0,n. Basc sequence s also called sequences of bnomal type, whch has generatng functon of the form b n (x) tn = exf(t), (2.7) n=0 6

7 where f(t) s the compostonal nverse of B(x) = x + b 2 x 2 + b 3 x 3 +. Sheffer sequences have generatng functons of the form n=0 s n (x) tn = 1 s(t) exf(t), (2.8) where f(t) s as above, and s(t) = n 0 s n(0)t n s a formal power seres of order 0. Substtutng B(t) for t n (2.8), we obtan the Appell relaton e xt = n=0 s n (x) s(b(t))[b(t)]n. From ths we conclude that Sheffer polynomals can be vewed as sequences of polynomals borthogonal to operator sequences of the form ϕ s (D) = s(b(d))[b(d)] n, where B(t), s(t) are formal power seres wth s(0) 0, B(0) = 0 and B (0) 0. It s known that Sheffer sequences wth specal ntal values can be used to study lattce path enumeraton and emprcal dstrbuton functons, where the correspondng delta operators are D and the backward dfference operator. See [5, 6] and ther references. For example, n studyng the order statstcs of a set of unformly dstrbuted random varables n [0, 1], let s n (x) := g n (x; a n, a n 1,..., a 1 ). Snce Ds n (x) = ns n 1 (x), we get a Sheffer sequence {s n } wth ntal values s n (a n ) = δ 0,n. Hence computng the emprcal dstrbuton s reduced to compute Sheffer polynomals wth gven ntal values. For lattce path enumeraton, one just replace D wth, (See Secton 3 and 4 for detals). Nederhausen has used Umbral Calculus to fnd explct solutons for lattces paths n the followng cases: (1) the boundary ponts a n are pecewse affne n n, (2) the steps are n several drectons, and (3) lattce paths are weghted by the number of left turns. The Sheffer sequence s also used to enumerate lattce paths nsde a band parallel to the dagonal, whch s a specal case descrbed n Secton 7. In ths paper, we use the framework of sequences of borthogonal polynomals for the followng reason: (1) It s more general, whle almost all the nce formulas for Sheffer sequences can be extended to ths general settng, (2) It s a natural algebrac correspondence for workng wth parkng functons and lattce paths, by the combnatoral decomposton theorem for parkng functons [1, Theorem5.1], and ts analog n lattce paths (c.f. Secton 4). And (3). The theory of borthogonal polynomals gves a unfed treatment to several combnatoral objects smultaneously, ncludng parkng functons, order statstcs of a set of unformly dstrbuted random varables, lattce paths, and the area-enumerator of lattce paths. 7

8 3 Dfference Goncarov Polynomal In ths secton we dscuss the dfference analog of Goncarov polynomals, whch s the sequence of polynomals borthogonal to a sequence of lnear operators defned by formal power seres of the (backward) dfference operators. Explctly, let p(x) be a polynomal n the vector space P = F [x]. Defne p(x) = p(x) p(x 1). Note that p(x) s a polynomal of x whose degree s one less than that of p(x). We follow the conventon that the upper factoral x (n) s x(x + 1) (x + n 1). Observe that the polynomals p n (x) = x (n) form a bass of the vector space P; p n (x) = np n 1 (x); and p n (x) x=0 = 0, whenever < n. Gven a sequence b 0, b 1,..., let ψ S ( ), s = 0, 1, 2,... be the lnear operators The dfference Goncarov polynomals ψ s ( ) = r=0 b (r) s r! r+s. (3.1) g n (x; b 0,..., b n 1 ), n = 0, 1, 2,... s the the unque sequence of polynomals satsfyng deg( g n (x; b 0,..., b n 1 )) = n and ψ s ( ) g n (x; b 0,..., b n 1 ) x=0 = δ sn. Many propertes of Goncarov polynomals have a dfference analog. whch are lsted n the followng lst. Most proofs are smlar to that of the dfferental case, and hence omtted or only gven a sketch. 1. Determnantal formula. g n (x; b 0,..., b n 1 ) = b 1 b (2) 0 b (3) 0 0 2! 3! b 0 1 b (2) 1 1 2! b (n 1) 0 (n 1)! b (n 2) 1 (n 2)! b (n 3) 2 b (n) 0 b (n 1) 1 (n 1)! b (n 1) 2 (n 2)! b (n 3)! b n 1 1 x x (2) 2! x (3) 3! x (n 1) (n 1)! x (n). (3.2) 8

9 2. Expanson formula. If p(x) s a polynomal of degree n, then p(x) = ψ ( )(p(x)) x=0 g (x; b 0,..., b 1 ). (3.3)! It s obtaned by applyng D on both sdes and then settng x = Lnear recurrence. Let p(x) = x (n) n (3.3), we get x (n) = b (n ) g (x; b 0,..., b 1 ). (3.4) 4. Appell relaton. 5. Dfference relaton. (1 t) x = t n g n (x; b 0,..., b n 1 ) (1 t) bn. n=0 g n (x; b 0,..., b n 1 ) = n g n 1 (x; b 1,..., b n 1 ), (3.5) and g n (b 0 ; b 0,..., b n 1 ) = δ 0n. (3.6) The above dfference relaton and ntal condton unquely determne the sequence of dfference Goncarov polynomals. 6. Summaton formula. When x, b 0 are ntegers, solvng the dfference relaton, we have the summaton x g n (x; b 0, b 1,..., b n 1 ) = n g n 1 (t; b 1,..., b n 1 ). (3.7) t=b 0 +1 Iteratng ths when x, b Z, we have the summaton formula g n (x; b 0,..., b n 1 ) = x 1 1 =b =b 1 +1 n 1 n=b n , (3.8) where w 2 =w 1 α() = α(w 1 ) + α(w 1 + 1) + + α(w 2 ) f w 1 w 2 ; 0 f w 1 = w 2 + 1; α(w 2 + 1) α(w 2 + 2) α(w 1 1) f w 1 > w (3.9)

10 7. Shft-nvarant formula. Usng a change of varable, the summaton relaton (3.7), and nducton, one obtan the followng shft-nvarant formula g n (x + t; b 0 + t,..., b n 1 + t) = g n (x; b 0,..., b n 1 ). (3.10) Note that Equaton (3.10) holds for all x, t, and b s, snce t s a polynomal dentty whch s true for nfntely many values of x, t and b s. 8. Perturbaton formula. g n (x; b 0,..., b m 1, b m + δ m, b m+1,..., b n 1 ) = g n (x; b 0,..., b m 1, b m, b m+1,..., b n 1 ) (3.11) g n m (b m + δ m ; b m, b m+1,..., b n 1 ) g m (x; b 0,..., b m 1 ). m Applyng the perturbaton formula repeatly, we get g n (x; b 0 + δ 0, b 1 + δ 1,..., b n 1 + δ n 1 ) = g n (x; b 0,..., b n 1 ) (3.12) g n (b + δ ; b,..., b n 1 ) g (x; b 0 + δ 0,..., b 1 + δ 1 ). 9. Bnomal expanson. If we expand g n (x + y; b 0,..., b n 1 ) usng the bass {x (n) }, we can get g n (x + y; b 0,..., b n 1 ) = g n (y; b,..., b n 1 )x (). (3.13) To see ths, note that (x + y) () = (x + y) ( 1), and g n (x + y; b 0,..., b n 1 ) = n g n 1 (x + y; b 0,..., b n 1 ). Now apply to both sde of Equaton (3.13) and set x = 0. Equaton (3.13) follows from nducton. Dfference Goncarov polynomal of low degrees can be easly computed by the determnant formula or the summaton formula. For example, g 0 (y) = 1, g 1 (y; b 0 ) = y (1) b (1) 0, g 2 (y; b 0, b 1 ) = y (2) 2b (1) 1 y(1) + 2b (1) 0 b(1) 1 b (2) 0, g 3 (y; b 0, b 1, b 2 ) = y (3) 3b (1) 2 y(2) + (6b (1) 1 b(1) 2 3b (2) 1 )y(1) b (3) 0 + 3b (2) 0 b(1) 2 6b (1) 0 b(1) 1 b(1) 2 + 3b (1) 0 b(2) 1. 10

11 In the followng specal cases, dfference Goncarov polynomals have a nce closed-form expresson. Case 1 b = b for all. Then g n (x, b,..., b) = (x b) (n). Case 2 b = y + ( 1)b forms an arthmetc progresson. Then we have the dfference analog of Abel polynomals: { (x y)(x y nb + 1) (n 1) n > 0; g n (x, y, y + b,..., y + (n 1)b) = 1 n = 0. To see ths, verfy the dfference relaton that g n (x; b 0,..., b n 1 ) = n g n 1 (x; b 1,..., b n 1 ) and g n (b 0 ; b 0,..., b n 1 ) = δ 0n. In partcular, g n (0; 1,..., n) = n+1( 2n n ) = Cn, where C n = 1 n+1( 2n n ) s the famous Catalan number. 4 Dfference Goncarov Polynomals and Lattce Paths In ths secton we descrbe a combnatoral decomposton whch allows us to relate the dfference Goncarov polynomals wth certan lattce paths n plane. Let x, n be postve ntegers. Consder lattces paths from (0, 0) to (x 1, n) wth steps (1, 0) or (0, 1). Denote by the sequence (x 0,..., x n ) such a path whose rght-most pont on the -th row s (x, ). Obvously, we always have x n = x 1. Gven b 0 b 1 b n 1 x, let LP n (b 0,..., b n 1 ) be the number of paths (x 0,..., x n 1 ) from (0, 0) to (x 1, n) wth steps (1, 0) and (0, 1) such that x < b for 0 n 1. s It s well-known that the total number of the paths from (0, 0) to (x 1, n) n the grd (x 1) n ( ) x + n 1 LP n (x,..., x) = = x(n) n. Another way of countng paths n the grd (x 1) n s to decompose the paths nto several classes as follows. Let (x 0,..., x n ) be such a path and be the frst row that x b. Each of such paths conssts of three parts: the frst part s a path from (0, 0) to (b 1, ) that never touches the ponts (b j, j) for j = 0, 1,...,, the second path conssts of one step (1, 0), from (b 1, ) to (b, ), and the thrd part s a path that goes from (b, ) to (x 1, n). The number of paths of the frst knd s LP (b 0,..., b 1 ), whle that of the thrd knd s ( x 1 b ) +n = (x b ) (n ). Therefore the total number of paths s n LP (b 0,..., b 1 ) (x b ) (n ). (n )! 11 (n )!

12 So x (n) = LP (b 0,..., b 1 ) (x a ) (n ). (4.1) (n )! Comparng Equatons (3.4) and (4.1), we get Theorem 4.1 LP (b 0,..., b 1 ) = 1! g (x; x b 0,..., x b n 1 ). In partcular, LP n (b 0,..., b n 1 ) = 1 g n(0; b 0,..., b n 1 ). (4.2) Usng the dentty ( x) (n) = ( 1) n x(x 1)(x 2) (x n + 1) = ( 1) n x (n) where x (n) s the lower factoral, and the determnant formula for g n, we get [( )] b LP n (b 0,..., b n 1 ) = det j ,j n 1 An equvalent descrpton for LP n (b 0,..., b n 1 ) s the number of nteger ponts n certan n- dmensonal polytope consdered by Ptman and Stanley n [8]. Let Π n (x) := {y R n : y 0 and y 1 + y y x x for all 1 n}, Ptman and Stanley computed the number of nteger ponts n the polytope Π n when x 1,..., x n are postve ntegers, and gave the formula where K n = {k N n : N(Π n (x)) = (x 1 + 1)(k1) k 1! k K n n =2 x (k ) k 2!, k j for all 1 n 1 and =1 k = n}. Lettng b 0 = x 0 + 1, b = 1 + j=0 x. Every nteger pont y = (y 1,..., y n ) Π n (x) corresponds unquely to a lattce path 0 r 0 r 1 r n 1 where r = y < b for all. Hence [( N(Π n (x)) = LP n (b 0, b 1,..., b n 1 ) = det 12 b j + 1 )] =1. (4.3) 0,j n 1

13 Formula (4.3) can also be derved from the Steck formula (c.f. Theorem 7.3) on the number of lattce paths lyng between two gven ncreasng sequences [11, 12]. The detaled can be found n the monograph [4] and correspondng theory of borthogonal polynomals are presented n Secton 7. As an applcaton of (4.2), let b =. Then LP n (1, 2,..., n) counts the number of Dyck paths. We have LP n (1,..., n) = 1 g n(0; 1,..., n) = n+1( 1 2n ) n, agan obtan the famous Catalan number. In general, we can consder the number of lattce paths from (0, 0) to (r + µn, n) (r, µ P), whch never touch the lne x = r + µy. Ths s just the case where b = r + ( 1)µ, and the number s gven by LP n (r, r + µ, r + 2µ,..., r + (n 1)µ) = 1 g n(0; r, r µ,..., r (n 1)µ) = 1 r(r + nµ + 1)(n 1) ( ) r r + n(µ + 1) =, (4.4) r + n(µ + 1) n a well-known result. (See, for example, [4, p.9]. In partcular, for r = 1 and µ = k, t counts the number of lattce paths from ( the orgn to (kn, n) that never pass below the lne y = x/k. The 1 formula (4.4) becomes (k+1)n ) kn+1 n, the nth k-catalan number [10, p. 175]. We can renterpret the perturbaton formula (3.12) usng paths. Gven two paths (a 0,..., a n 1 ) and (c 0, c 1,..., c n 1 ) wth a c, We consder all paths that never touch (c 0, c 1,..., c n 1 ). Frst, t s LP n (c 0,..., c n 1 ) as defned. Secondly, we can also count the paths that never touch (c 0,..., c n 1 ), whle they touch the path (a 0,..., a n ) on -th row for the frst tme. The total number of such paths s LP (a 0,..., a 1 )LP n (c a, c +1 a,..., c n 1 a ). So we have the formula LP n (c 0,..., c n 1 ) = LP (a 0,..., a 1 )LP n (c a, c +1 a,..., c n 1 a ) (4.5) +LP n (a 0,..., a n ). Convertng the equaton usng dfference Goncarov polynomals, we have g n (x; x a 0, x a 1,..., x a n 1 ) = g n (x; x c 0,..., x c n 1 ) g n (0; a c,..., a c n 1 ) g (x; x a 0,..., x a 1 ). Replacng x c wth b, c a wth δ, and usng the shft formula (3.10) on 13

14 g n (0; a c,..., a c n 1 ) by g n (0; a c,..., a c n 1 ) = g n (0; δ, b +1 b δ,..., b n 1 b δ ) we get the perturbaton formula (3.12) agan. = g n (b + δ ; b, b +1,..., b n 1 ), Remark. Let b 0 b 1 b n 1 be a sequence of ntegers. Denote by LP < n the set of nteger sequences (r 0 < r 1 < < r n 1 ) such that 0 r < b for = 0, 1,..., n 1. Then LP n <, the cardnalty of LP < n, can be obtaned as follows: Let s = r ( 1). Then s 0 s 1 s n 1 and 0 s < b ( 1). Hence LP n < (b 0,..., b n 1 ) = LP n (b 0, b 1 1,..., b n 1 n 1). Alternatvely, we can use the forward dfference operator f and ts basc polynomals x (n) = x(x 1)... (x n + 1) to replace and x (n) n (3.1), where f p(x) = p(x + 1) p(x). Explctly, let ψ S ( f ) = (b s) (r) r=0 r! r+s f. Denote the correspondng sequence of borthogonal polynomals by g f,n (x; b 0,..., b n 1 ). The determnant formula of g f,n (x; b 0,..., b n 1 ) s obtaned from (3.2) by replacng each upper factoral a () wth the lower factoral a () = a(a 1)... (a + 1). Under ths settng, we have LP n < (b 0,..., b n 1 ) = 1! g f,n(0; b 0,..., b n 1 ). The above two approaches yeld the followng determnant formulas for LP n < (b 0,..., b n 1 ). [( )] [( )] LP n < b b + j (b 0,..., b n 1 ) = det = det j + 1 j + 1 0,j<n. 0,j<n 5 q-goncarov Polynomal For u-parkng functons, the sum enumerator S n (q, u) = (a 1,...,a n) qa 1+a 2 + +a n, where (a 1,..., a n ) ranges over all u-parkng functons, s just the specalzaton of the (dfferental) Goncarov polynomals where u s replaced wth 1 + q + + q u 1. Ths s not the case for lattce paths and dfference Goncarov polynomals. Defne the area-enumerator of lattce paths to be Area n (q; b) := q x 0+x 1 + +x n 1, (5.1) (x 0,...,x n 1 ) LP n(b) where LP n (b) s the set of lattce paths from (0, 0) to (x 1, n) (x 1 b n 1 ) that never touch (b 0, b 1,..., b n 1 ). Note that x 0 + x x n 1 s the area of the regon bounded by the path and the lnes x = 0 and y = n. To study Area n (q; b), we develop the q-analog of dfference Goncarov polynomals. We use the followng the conventons that (n) q = 1 qn 1 q ; (n) q! = (1) q (n) q ; and the rsng q-factoral { (1 A) (1 Aq (A; q) n = n 1 ) f n > 0, 1 f n = 0. 14

15 Let p(y) be a polynomal n the rng F (q)[y]. Defne q p(y) = p(y) p(y/q) (1 q)y/q. It s easy to check that q p(y) s a polynomal of y whose degree s one less than that of p(y). Observe that the polynomals p n (y) = (y; q) n form a bass of the rng F (q)[y]; q p n (y) = (n) q p n 1 (y); and qp n (y) y=1 = 0, whenever < n. Let ψ q,s ( q ), s = 0, 1, 2,..., be the sequence of lnear operators ψ q,s (D q ) = s=0 (b s ; q) r r+s q, (5.2) (r) q! and defne the dfference q-goncarov polynomals g n (q; y; b) = g n (q; y; b 0,..., b n 1 ) to be the sequence of polynomals borthogonal to ψ q,s ( q ),.e., ψ q,s ( q )g n (y; b; q) y=1 = (n) q!δ sn. Smlar propertes satsfed by the regular Goncarov polynomals can be generalzed to a q-analog for the dfference q-goncarov polynomals. We lst the man results n the followng. 1. Determnantal formula. g n (q; y; b 0,..., b n 1 ) = (n) q! 1 (b 0 ; q) 1 (b 0 ;q) 2 (2) q! (b 0 ;q) 3 (3) q! 0 1 b 1 (b 1 ;q) 2 (2) q! b 2 (b 0 ;q) n 1 (n 1) q! (b 1 ;q) n 2 (n 2) q! (b 2 ;q) n 3 (n 3) q! (b 0 ;q) n (n) q! (b 1 ;q) n 1 (n 1) q! (b 2 ;q) n 1 (n 2) q! b n 1 1 (y; q) 1 (y;q) 2 (2) q! (y;q) 3 (3) q! (y;q) n 1 (n 1) q! (y;q) n (n) q!. 2. Expanson formula. For any polynomal p(y) F (q)[y], p(y) = To verfy, apply D on both sdes and then set y = 1. ψ q, ( q )(p(y)) y=1 g n (q; y; b 0,..., b 1 ), (5.3) () q! 3. Lnear recurson. (y; q) n = (b ; q) n g (q; y; b 0,..., b 1 ). (5.4) q 15

16 4. Appell Relaton. Snce n (a; q) n (q; q) n t n = (at; q) (t; q), where (a; q) = k=0 (1 aqk ), we have the generatng functon (yt; q) = g (q; y; b 0,..., b 1 ) (b t; q) t () q! 5. Dfference relaton. wth the ntal condtons q g n (q; y; b 0,..., b n 1 ) = (n) q g n 1 (q; y; b 1,..., b n 1 ), (5.5) g n (q; b 0 ; b 0,..., b n 1 ) = δ 0n. (5.6) 6. Bnomal expanson. The bnomal expanson of Goncarov polynomals becomes g n (q; ty; b 0,..., b n 1 ; q) = t g n (q; t; b,..., b n 1 )(y; q). (5.7) q Ths s because q (ty; q) = () q t(ty; q) 1, and q g n (q; ty; b 0,..., b n 1 ) = (n) q tg n 1 (q; ty; b 1,..., b n 1 ). Now apply q to both sde of the equaton and set y = Summaton formula. Let y = q x and b = q a, where x and a are ntegers, we have x g n (q; y; b 0,..., b n 1 ) = (1 q) q 1 (n) q g n 1 (q; q ; b 1,..., b n 1 ), =a 0 +1 where the sum s defned the same as n 3.9. Ths s because g n (q; y; b 0,..., b n 1 ) = g n (q; q x 1 ; b 0,..., b n 1 ) + (1 q)q x 1 (n) q g n 1 (q; q x ; b 1,..., b n 1 ) and the ntal condton (5.6). sum formula Iterate t we obtaned the g n (q; y; b 0,..., b n 1 ) = (1 q) n (n) q! x q q 2 1 From ths we derve the shft formula: 1 =a =a 1 +1 n 1 n=a n 1 +1 q n 1.(5.8) g n (q; yq ξ ; b 0 q ξ,..., b n 1 q ξ ) = q nξ g n (q; y; b 0,..., b n 1 ). (5.9) Snce g n (q; y; b) s a polynomal of y and b s over a feld and the equaton above holds for nfntely many y s and b s, t holds for all y and b s. 16

17 Examples of the q-goncarov polynomals follow. g 0 (q; y) = 1, g 1 (q; y; b 0 ) = (y; q) 1 (b 0 ; q) 1, g 2 (q; y; b 0, b 1 ) = (y; q) 2 (2) q!(b 1 ; q) 1 (y; q) 1 + (2) q!(b 0 ; q) 1 (b 1 ; q) 1 (b 0 ; q) 2, g 3 (q; y; b 0, b 1, b 2 ) = (y; q) 3 (3) q (b 2 ; q) 1 (y; q) 2 + ((3) q!(b 1 ; q) 1 (b 2 ; q) 1 (3) q (b 1 ; q) 2 )(y; q) 1 (b 0 ; q) 3 + (3) q (b 0 ; q) 2 (b 2 ; q) 1 (3) q!(b 0 ; q) 1 (b 1 ; q) 1 (b 2 ; q) 1 + (3) q (b 0 ; q) 1 (b 1 ; q) 2. In partcular, n some specal cases we have nce closed formula, g n (q; q x ; q b,..., q b ) = q nb (q x b ; q) n and g n (q; q x ; q y, q y+1,..., q y+n 1 ) = ( 1) n q (n 2) nx (q y x ; q) n. 6 q-goncarov Polynomals and Area of Lattce Paths Defne the q-upper factoral x (n) q = (1 q x ) (1 q x+n 1 )/(1 q) n. In ths secton we use the dfference q-goncarov polynomal developed n the prevous secton to represent the areaenumerator of lattce paths wth upper constrant. Consder an (x 1) by n grd consstng of vertcal and horzontal lnes. Let (x 0,..., x n 1 ) be a path that goes from (0, 0) to (x 1, n) along the grd, whle the rght-most pont on the -th row s (x, ) for = 0, 1,..., n 1. Gven a path (b 0, b 1,..., b n 1 ) wth b 0 b 1 b n 1 x 1, let Area n (q; b) be gven n Formula (5.1). We establsh a recurrence of Area n (q; b) by computng the area-enumerator of Area n (q; x 1, x 1,..., x 1) n two ways. Frst, t s well-known that the area-enumerators of all the paths n the grd (x 1) n s ( ) x + n 1 Area n (q; x 1,..., x 1) = n q = x(n) q (n) q!. Now apply the decomposton as n 4. Let (x 0,..., x n ) be a path n (x 1) n for whch be the frst row that the path touches the path (b 0,..., b n 1 ),.e., x b. Each of such paths conssts of two parts: the frst part s a path from (0, 0) to (b 1, ) that never touches the path (b 0,..., b 1 ), the second part conssts of one horzontal step from (b 1, ) to (b, ), and the thrd part s a path that goes from (b, ) to (x 1, n). The area contrbuted by the frst part s Area (q; b 0,..., b 1 ), whle 17

18 that of the second knd and the thrd s q b (n ) ( x 1 b + n n area-enumerator of all the paths s So whch leads to (q x ; q) n = x (n) q = Area (q; b 0,..., b )q b (n ) (x b ) (n ) q. (n ) q! )q = qb (n ) (x b ) (n ) q (n ) q!. Therefore the (n) q! (n ) q! Area (q; b 0,..., b )q (n )b (x b ) (n ) q, (6.1) (n) q! [ ] (n ) q! (1 q) (q x b ; q) n q (n )b Area (q; b 0,..., b ). (6.2) Comparng equatons (5.4) and (6.2), we need to make the power of q on the rght-hand sde of equaton (6.2) dependng only on and b. To acheve ths, we replace q by 1 q n equaton (6.2). Then (q x ; q) n becomes ( 1) n q (nx+(n 2)) (q x ; q) n, and (n) q! becomes q (n 2) (n)q!. Substtutng nto (6.2) and reorganzng the equaton, we get Then (q x ; q) n = ( n ) Comparng equatons (5.4) and (6.3), we obtan Area ( 1 q ; b 0,..., b ) = [ ] () q!( 1) (q x b ; q) n q x (1 1 q q ) Area ( 1 q ; b 0,..., b ). (6.3) ( 1) () q!q x (1 1 q ) g n(q; q x ; q x b 0,..., q x b 1 ). Let f(q; x, b 0,..., b n 1 ) be a polynomals of q wth parameters x, b 0,..., b n 1 gven by f(q; x, b 0,..., b n 1 ) = g n (q; q x ; q x b 0,..., q x b 1 ). Theorem 6.1 The area-enumerators of lattce paths n the rectangle (x 1) n that stays strctly above the path (b 0,..., b n ) s Area n (q; b) = = ( 1) n (1 q) n (n) q! q n(n 1) 2 +nx f( 1 q ; x, b 0,..., b n 1 ) ( 1) n (1 q) n (n) q! q n(n 1) 2 f n ( 1 q ; 0, b 0,..., b n 1 ). (6.4) 18

19 7 Two-Boundary Extensons Goncarov polynomals can be extended to represent parkng functons and lattce paths wth both both upper and lower boundares. 7.1 Parkng functons wth two-sded boundary u-parkng functons are nteger sequences whose order statstcs s bounded by a prefxed sequence u from above. We may consder parkng functons wth both upper and lower constrants. More precsely, let r 1 r 2 r n and s 1 s 2 s n be two sequence of non-decreasng ntegers. A (r, s)-parkng functon of length n s a sequence (x 1,..., x n ) whose order statstcs satsfy r x () < s. Denoted by P n (r, s) = P n (r 1,..., r n ; s 1,..., s n ) the number of (r, s)-parkng functons of length n. The formula P n (r, s) can also be expressed as borthogonal polynomals. Let (a 0, a 1, a 2,... ) and (b 0, b 1, b 2,... ) be two sequences of numbers. Goncarov polynomals Defne the extended g n(x; a, b) = g n(x; a 0, a 2,..., a n 1 ; b 0, b 1,..., b n 1 ), n = 0, 1, 2,..., to be the sequence of polynomals borthogonal to the operators ϕ s (D) = D s (b s a s+r 1 ) r +D r, (7.1) r! r=0 where x + = max(x, 0). (Here we set a 1 = 0.) By the determnant formula (2.3), (b 0 a 1 ) 1 (b 0 a 0 ) 2 + (b 0 a 2 ) 3 + (b 0 a n 2 ) + 2! 3!... n 1 + (b 0 a n 1 ) n + (n 1)! (b 1 a 2 ) 0 1 (b 1 a 1 ) 2 + (b 1 a n 2 ) + 2!... n 2 + (b 1 a n 1 ) n 1 + (n 2)! (n 1)! g n(x; (b 2 a n 2 ) a, b) = (b 2 a 2 ) +... n 3 + (b 2 a n 1 ) n 2 + (n 3)! (n 2)! (b n 1 a n 1 ) + x 1 x 2 x 3 x 2! 3!... n 1 x n (n 1)!. In partcular, g n(0; a, b)) = ( 1) n det[(b a j ) j +1 + /(j +1)!]. By the lnear recurrence equaton (2.5), we have x n = (b a n 1 ) + n g (x; a, b). 19

20 It follows that for n 1, (b a n 1 ) + n g (0; a, b) = 0. (7.2) The sequence {g n(0, a, b) s unquely determned by the above recurrence and ntal values g 0 (0; a, b) = 1, g 1 (0; a, b) = (b 0 a 0 ) +. Theorem 7.1 P n (r 1,..., r n ; s 1,..., s n ) = ( 1) n g n(0; r 1,..., r n ; s 1,..., s n ). (7.3) To prove Theorem 7.1, t s suffcent to show that ( 1) (s +1 r n ) n + P (r, s) = 0, (7.4) for n 0, and P 1 (r, s) = (s 1 r 1 ) +. The ntal value s clear. In the followng we gve two proof of equaton (7.4). The frst one s based on a weghted verson of ncluson-excluson prncple. The second s an nvoluton on the set of marked parkng functons, whch reveals some ntrnsc structures of two-sded parkng functons. Frst Proof of (7.4). Let M(S) be the set of all sequences α of length n such that α S s a (r, s)-parkng functon of length S, and each term n α S c les n [r n, s +1 ), where S c = [n] \ S. Then (7.4) s equvalent to S ( 1) S M(S) = 0, where the sum ranges over all subsets S [n]. For any sequence α, let T [α] = {S : α M(S)}. It s suffcent to show that ( 1) T = 0. (7.5) T T [α] Observe that f α M(S) and S S, then α M(S ). Hence T [α] s a flter n the power set of [n]. T [α] f and only f α s a (r, s)-parkng functon. When T [α], let S 1,..., S r be the mnmal elements of T [α]. S 1,..., S r satsfy the followng propertes. 1. S < n. For any (r, s)-parkng functon α, deletng the largest element whch s n [r n, s n ), the remanng s a (r, s)-parkng functon of length n 1. Hence T [α] {[n]}. 2. S 1 = S 2 = = S r = k for some k < n. Assume k = S 1 < S 2 = l. The condton α M(S 1 ) mples all terms of α are less than s k+1, and at least n k of them are larger than or equal to r n. In partcular, the largest element n α S2 les n [r n, s k+1 ). Then S 2 s not mnmal. 20

21 3. S 1 S 2 S r [n]. Otherwse, every term n α appears n some (r, s)-parkng functon of length k < n, and hence less than s k. And any term n a poston of S 1 \ S 2 s greater than or equal to r n. Hence the mnmal element of T [α] has length S 1 1, a contradcton. Denoted by F(S 1,..., S r ) the flter of the power set of [n] generated by S 1,..., S r, and W (F(S 1,..., S r )) = w(t ), T F(S 1,...,S r) for a weght functon w(t ). Note that F(S 1,... S r ) = F(S 1 )... F(S r ), and F(S ) F(S j ) = F(S S j ), usng Incluson-Excluson, we have W (F(S 1,..., S r )) = W (F(S )) W (F(S S j )) + W (F(S S j S k )). <j <j<k Lettng the weght w(t ) = ( 1) T. Formula (7.5) follows from the equaton W (F(S)) = ( 1) n S = 0 whenever S [n]. Second Proof of (7.4). ( n even We gve a bjectve proof for the equvalent form ) (s +1 r n ) n + P (r, s) = odd ( n ) (s +1 r n ) n + P (r, s). (7.6) The left-hand sde of (7.6) s the cardnalty of the set M of pars (α, S) where α s a sequence of length n, S [n] wth S even, such that α S s a (r, s)-parkng functon of length S, and any term n α S c les n [r n, s S +1 ). The rght-hand sde of (7.6) s the cardnalty of the set N of pars (α, S) where (α, S) s smlar as those appeared n M, except that S beng odd. For a sequence α, let m = max(α) be the frst maxmal entry of α. Let pos(m) be the poston of m. Defne σ : M N by lettng σ(α, S) = (α, T ) where { (α, S \ {pos(m)}), f pos(m) S, T = (α, S {pos(m)}), f pos(m) / S. The map σ s well-defned: For any par (α, S) wth S even, clearly T s odd. Case 1. If pos(m) S, then deletng m from the subsequence n S, we obtan a (r, s)-parkng functon of length S 1 = T. The condton that m = max(α) and m [r S, s S ) mples that for any term x n α T c, x m < s S = s T +1. In addton, f S c, then m x r n for any x S c ; f S c =, then α tself s a (r, s)-parkng functon of length n, hence m r n. Ths proves that n the case pos(m) S, (α, T ) N. Case 2. If pos(m) / S, then any term x S c les n [r n, s S +1 ) [r n, s S +2 ). As m [r n, s S +1 ), jonng m to the subsequence on S wll result n a (r, s)-parkng functon of length S + 1 = T. In both cases, σ maps a par n M to a par n N. 21

22 It s easly seen that σ has the nverse map σ 1 (α, T ) = (α, S) where { (α, T \ {pos(m)}), f pos(m) T, S = (α, T {pos(m)}), f pos(m) / T. Ths proved that σ s a bjecton from M to N. Equaton (7.1) should be compared wth followng formula of Steck [11, 12] for the cumulatve dstrbuton functon of the random vector of order statstcs of n ndependent random vaables wth unform dstrbuton on an nterval. Let 0 r 1 r 2 r n 1 0 s 1 s 2 s n 1, be gven constants such that r < v for = 1, 2,..., n. If X (1), X (2),..., X (n) are the order statstcs n ascendng order from a sample of n ndependent unform random varables wth ranges 0 to 1, then P r(r X () s, 1 n) = det[(s r j ) j +1 + /(j + 1)!]. (7.7) The dfference between equatons (7.1) and (7.7) s that n a (r, s)-parkng functon, the sequence can only assume nteger values. Whle a unform random varable n [0, 1] corresponds to realvalued parkng functons, [2]. Hence equaton (7.1) can be vewed as a dscrete extenson of the Steck formula (7.7). The equaton (7.4) can be extended to the sum-enumerator of (r, s)-parkng functons. Defne S n (q; r, s) = α=(a 1,...,a n) q a 1+ +a n where the sum ranges over all (r, s)-parkng functons of length n. Wth a smlar proof to that of (7.4), we can show that ( 1) ((s +1 ) q (r n ) q ) n S (q; r, s) = 0, where the factor s +1 ) q (r n ) q s 0 f r n s +1. Hence, the sum-enumerator s a specalzaton of the polynomal P n (r, s): Theorem 7.2 where and S n (q; r, s) = P n (r(q), s(q)), r(q) = ((r 1 ) q, (r 2 ) q,..., (r n ) q ), s(q) = ((s 1 ) q, (s 2 ) q,..., (s n ) q ). 22

23 7.2 Lattce paths wth two-sded boundary The number of lattce paths wth two boundares was obtaned as a determnant formula by Steck [11, 12]. Such enumeraton and varous generalzatons has been extensvely studed, for example, n [4, Chapter2]. Hence we just lst the man results on the subject, and explan the connecton to borthogonal polynomals. Theorem 7.3 (Steck) Let a 0 a 1 a m and b 0 b 1 b m be sequences of ntegers such that a, b. The number of sets of ntegers (r 0, r 1,..., r m ) such that r 0 < r 1 < < r m and a < r < b for 0 m s the (m + 1)-th determnant det(d j ) where d j = ( b a j +j 1) j +1 f j and b a j > 1. Otherwse d j = 0. Denoted by LP n (a, b) the number of lattce paths (x 0, x 1,..., x n 1 ) from (0, 0) to (x 1, n) satsfyng a x < b < x. Steck s formula gves [( )] (b a j ) + LP n (a, b) = det. (7.8) j + 1 Formula (7.8) s a specalzaton of extended dfference Goncarov polynomals. Gven two sequences a = (a 0, a 1, a 2,... ) and b = (b 0, b 1, b 2,... ), let g n(x; a, b) = g n(x; a 0, a 1,..., a n 1 ; b 0, b 1,..., b n 1 ) (n = 0, 1, 2... ) be the sequence of polynomals borthogonal to the operators ( ) ψ S ( ) = s ( 1) r (bs a s+r 1 ) + r. (7.9) r Then r=0 ( ) g n(0; (b a j ) + a, b)) = det[ ] = LP n (a, b). (7.10) j + 1 These equatons enable us to enumerate LP n (a, b) from the theory of borthogonal polynomals. Snce generally, recurrence relatons and generatng functons are major technques to solve a countng problem, we show how such results on LP n (a, b) follow from the propertes of g n(0; a, b)). Frst, the lnear recurrence (3.4) becomes x (n) =! ( 1)n ( (b a n 1 ) + n ) g (x; a, b) It follows that ( ) δ 0,n = ( 1) (b a n 1 ) + 1 ( ) n! g (0; a, b) = ( 1) (b a n 1 ) + LP (a, b). (7.11) n 23

24 Equaton (7.11) gves a lnear recurrence to compute LP n (a, b). Ths equaton has been obtaned n [4] as well, see equaton (2.37). One can also prove t combnatorally by countng alternatvely the set M of all pars (α, ) where α = (α 1, α 2,..., α n ) s an nteger sequence satsfyng (1) α 1 α 2 α, (2) a j α j < b j for each j = 0, 1,...,, and (3) α +1 < α +2 < < α n [a n, b +1 ) n. By a smlar argument, f one defne Area n (q; a, b) = x q x 0+x 1 + +x n 1 a 0 a n 1, Then δ 0,n = ( 1) ( (b a n 1 ) + n ) Area (q; a, b). q From the Appell relaton we get the dentty 1 (1 t) x = n=0 n=0 g n(x; a, b) ψ n(t), LP n (a, b) ψ n(t) where ψ n (t) s gven n (7.9). In partcular, when a = k + c, b = k + d wth c < d,.e., lattce paths are restrcted n a strp of wdth d c, ψ n (t) = t n f(t) where f(t) s a polynomals of degree. Hence the sequence LP n(a, b) has a ratonal generatng functon d c k n=0 = 1, LP n (a, b) tn = 1 f(t). It remans true even the ntal boundares a, b for = 0, 1,..., T are arbtrary. References 1. J. Kung and C. Yan, Goncarov polynomals and parkng functons, Journal of Combnatoral Theory, Seres A 102(2003), J. Kung and C. Yan, Expected Sums of Moments General Parkng Functons, Annals of Combnatorcs, vol. 7(2003), J. Kung and C. Yan, Exact Formula for Moments of Sums of Classcal Parkng Functons, Advances n Appled Mathematcs, vol 31(2003), 24

25 4. S. G. Mohanty, Lattce Path Countng and Applcatons, Academc Press, H. Nederhausen, Sheffer polynomals for computng exact Kolmogorov-Smrvov and Reny type dstrbutons. Annals of Statstcs 9(1981), H. Nederhausen, Lattce Path Enumeraton and Umbral Calculus, Advances n Combnatoral Methods and Applcaton s (edtor: N. Balakrshnan), Brkhuser Boston (1997). 7. E.J.G.Ptman, Smple proofs of Steck s determnantal expressons for probabltes n the Kolmogorov and Smrnov tests. Bull. Austral. Math. Soc., 7(1972), J. Ptman and R. Stanley, A polytope related to emprcal dstrbutons, plane trees, parkng functons, and the assocahedron, Dscrete Comput. Geom. 27(2002), no.4, G-C. Rota, D. Kahaner and A. Odlyzko, On the foundatons of combnatoral theory. VIII. Fnte operator calculus. J. Math. Anal. Appl. 42 (1973), R. Stanley, Enumeratve Combnatorcs, Volume 2. Cambrdge Unversty Press, G.P. Steck, The Smrnov two sample tests as rank tests, Ann. Math. Statst., 40: , G.P. Steck. Rectangle probabltes for unform order statstcs and the probablty that the emprcal dstrbuton functon les between two dstrbuton functons. Ann. Math. Statst., 42:1 11,