14 Classic Counting Problems in Combinatorics

Transcription

1 14 Classic Coutig Problems i Combiatorics (Herbert E. Müller, May 2017, herbert-mueller.ifo) Combiatorics is about coutig objects, or the umber of ways of doig somethig. I this article I preset some classic fuctio coutig problems. These iclude permutatios of a set, compositios ad partitios of a set or umber, ad (multi)arragemets ad (multi-)combiatios from a set. I will repeatedly refer to OEIS (oeis.org): the Olie Ecyclopedia of Iteger Sequeces. This article has two parts. I the first part we cosider selfmaps (N N) of a -set N. Startig with the problem of coutig all selfmaps, we create ew ad more iterestig problems i two ways: 1) Cout bijective selfmaps (permutatios) oly. 2) Itroduce a equivalece relatio by cojugatig the selfmaps with the permutatio 1 group: S (N N) S ; the cout the equivalece classes.* I this way we obtai 2 2=4 coutig problems ivolvig selfmaps of a set. I the secod part we cosider fuctios (K N) from a -set N to a k-set K. Startig with the problem of coutig all fuctios, there are agai two ways to create ew problems: 1) Cout surjective fuctios oly, or ijective fuctios oly. 2) Itroduce a equivalece relatio o the fuctios, ad cout the equivalece classes. This is doe with the permutatio group S actig o N, or with S k actig o K, or both.* I this way we obtai 3 4=12 coutig problems. 2 of these problems are trivial, ad we are left with 10 coutig problems ivolvig fuctios from oe set to aother. *The equivalece class of a fuctio f is the orbit of f uder the actio of a group. Part 1: Permutatios ad geeral selfmaps Here we cosider selfmaps (N N) Table: 2 2 selfmap coutig problems & their solutios. of a -set N ={a, b, c,...}. We cout all selfmaps, or bijective selfmaps (permutatios) oly, ad the selfmaps, or the equivalece classes obtaied by cojugatio with the permutatio group S. fuctio/class bijective all N N permutatios selfmaps! of a -set of a -set S 1 (N N) S partitios selfmap types P() of a umber of a -set A 1372 Permutatios of a set ad equivalece classes The permutatios of a -set N={a, b, c,...} form the symmetric group or permutatio group S. Example: ( a b c d a c d b) (a a,b c,c d,d b) is a permutatio of the 4-set {a,b,c,d}. A compact way of writig dow a permutatio is the cycle otatio, i our example (bcd)(a). (bcd) is a cycle of legth 3, meaig b c,c d,d b; (a) is a cycle of legth 1, meaig a a. The cycle legths listed i decreasig order form a partitio of the umber of elemets of, called the cycle type. The cycle type of the permutatio (bcd)(a) is 3+1, or (3,1). 1

2 Now let p k ad p l be 2 permutatios of N (1 k,l!). We say p k ad p l are equivalet if there is a permutatio p m such that p l = p m p k p m 1. The orbit of p k uder cojugatio with S is the subset of permutatios {p m p k p m 1 p m S }; it is a equivalece class of permutatios. It ca be show that equivalet p k ad p l belog to the same cycle type, ad that ay two permutatios with the same cycle type are equivalet. I other words, there is a 1-to-1 correspodece betwee the partitios p of the umber ad the equivalece classes of S. Examples The 1-set {a} has 1 permutatio, cycle type (1): (a). The 2-set {a,b} has 2 permutatios belogig to 2 cycle types: 1 (2)-permutatio (a,b) 1 (1,1)-permutatio (a)(b) The 3-set {a,b,c} has 6 permutatios belogig to 3 cycle types : 2 (3)-permutatios (a,b,c), (a,c,b) 3 (2,1)-permutatios (b,c)(a), (a,c)(b), (a,b)(c) 1 (1,1,1)-permutatio (a)(b)(c) The 4-set {a,b,c,d} has 24 permutatios belogig to 5 cycle types: 6 (4)-permutatios (abcd), (abdc), (acbd), (acdb), (adbc), (adcb) 8 (3,1)-permutatios (bcd)(a), (bdc)(a), (acd)(b), (adc)(b), (abd)(c), adb)(c), (abc)(d), (acb)(d) 3 (2,2)-permutatios (ab)(cd), (ac)(bd), (ad)(bc) 6 (2,1,1)-permutatios (ab)(c)(d), (ac)(b)(d), (ad)(b)(c), (bc)(a)(d), (bd)(a)(c), (cd)(a)(b) 1 (1,1,1,1)-permutatio (a)(b)(c)(d) Formulas The umber of permutatios of a -set is!= (1) : factorial. The umber of partitios of is the partitio-fuctio P() = A 41 (), with P(N) = (1,1,2,3,5,7,11,15,22,30,42...) (OEIS A000041, startig with P(0)=1). There is o fiite formula for P(), but a recursio formula due to Euler: P()=P(1)+P(2)P(5)P(7)+P(12)+P(15)P(22)P(26)... Remark 1: Although this is beyod the scope set for this article, let's also cout the fuctios i a give class.! The umber of type- p-permutatios of a -set is (p k m k ). The sum over all partitios is! { p} (p k m k ) =!. Here m is a vector with the same legth as the partitio p; it is defied by m 1 =1, m k+1=1+m k[p k+1=p k] (with [A]=0 if the statemet A is false, [A]=1 if A is true). Eg. if p=(5,4,4,4,2,1,1), the m=(1,1,2,3,1,1,2). Remark 2: Istead of the permutatios of a certai cycle type p, oe might also cout the permutatios with the same umber k of cycles. The umber of such k-permutatios of a -set is the 1 st usiged Stirlig umber. Geeral selfmaps of a set ad equivalece classes The geeral selfmap of a set, for example (a a,b d,c d,d b) ca be represeted by the familar two-row bracket, here ( a b c d a d d b), or just by the secod lie (a,d,d,b), or by a labeled ad orieted graph graph, here (labels letters, orietatio arrows). The graph is a rig with braches if the fuctio is coected, ad a collectio of rigs with braches otherwise. If we remove the letters i the example graph, we obtai. 2

3 If we further stipulate that the directio i the rig is couter-clockwise (ecessary for rigs with 3 or more vertices), we ca remove the arrows too (the directios i the braches are always towards the rig), ad obtai. This ulabeled graph (icludig the couterclockwise sese i the rig) is preserved uder cojugatio with S, ad I will call it the selfmap type. Selfmaps belogig to the same type form a equivalece class. If the selfmap type is a collectio of rigs (without braches), the fuctio is a permutatio. Examples The 1-set {a} has 1 selfmap: the idetity ( a a) (a). The selfmap type is. The 2-set {a,b} has 2 2 =4 self-maps belogig to 3 selfmap types: 1 2 ( a b a a) (a,a) 1 The 3-set {a,b,c} has 3 3 =27 selfmaps belogig to 7 selfmap types: 2 permutatios (b,c,a), (c,a,b) 6 selfmaps (b,c,b), (c,c,b), (b,a,a), (b,a,b), (c,a,a), (c,c,a) 6 selfmaps (b,c,c), (c,a,c), (c,b,b), (b,b,a), (a,c,a), (a,a,b) 3 selfmaps (b,b,b), (c,c,c), (a,a,a) 3 permutatios (b,a,c), (c,b,a), (a,c,b) 6 selfmaps (b,b,c), (a,a,c), (a,c,c), (a,b,b), (a,b,a), (c,b,c) 1 permutatio (a,b,c) The 4-set {a,b,c,d} has 4 4 =256 selfmaps belogig to 19 selfmap types (I give just those). Formulas The umber of selfmaps of a -set is. The umber of types of coected selfmaps is A 2861 (N*) = (1,2,4,9,20,51,125...). (This is the umber of -vertices rigs-with-braches; a chiral graph ad its mirror image must be couted separately.) OEIS A002861: "Number of coected fuctios (or mappig patters) o ulabeled poits, or umber of rigs ad braches with edges." The umber of selfmap types is A 1372 (N*) = (1,3,7,19,47,130,343...). A 2861 ( p k )+ m k 1 Euler trasform: A 1372 ()= { p} (k ). m k (Every -graph belogs to a partitio p of. The multiplicity of the partitio is m, see above. The formula is the sum over partitios of the umber of -graphs belogig to a partitio.) OEIS A001372: "Number of mappigs (or mappig patters) from poits to themselves; umber of edofuctios. Euler trasform of A " 3

4 Coutig the types of coected -selfmaps: The graphs are -vertices rigs-with-braches. First oe couts the umber of -graphs with k-rigs (k ), see o the right; the oe sums over k. OEIS A002861: "Number of coected fuctios (or mappig patters) o ulabeled poits, or umber of rigs ad braches with edges." See also A "Number of -ode coected graphs with exactly oe cycle of legth k [...] " ad A "Number of -ode coected graphs with oe cycle [...]." The etries A (6,4)=7 ad A 68051(6) =49 must be augmeted by 2, to accout for 2 mirror-pairs of chiral graphs. Triagular array: umber of coected -selfmap types with a k-rig. (,k) Part 2: Partitios, Compositios, Combiatios & Arragemets Here we cosider fuctios (K N) from a -set ={a, b, c,...} to a k-set K={1...k}. We cout all fuctios, or surjective fuctios oly, or ijective fuctios oly, ad the fuctios themselves, or equivalece classes obtaied by permutatio of N, or of K, or both. Table: 3 4 fuctio coutig problems ad their solutios. fuctio or class surjective all ijective (K... N) k-compositio of a -set (K... N) S k-compositio S k (K... N) of a umber k! { -multiarrage k} ~ from a k-set k -arragemet from a k-set! ( k ) ( 1 k1) -multicombia ~ from a k-set ( k+1 ) -combiatio from a k-set ( k ) k-partitio of a -set { k} k-partitio of a -set 1 { k} oly oe class S k (K... N) S k-partitio of a umber [ k] k-partitio of a umber [ +k k ] oly oe class Remarks: The (K N) fuctio coutig approach to combiatorics gives a somewhat surprisig, ad at first glace ot etirely coheret selectio of combiatorial problems. 1. Two problems are trivial. 2. The order i which the problems appear (say coutig from top to bottom ad the from left to right) is probably ot how you ad I would would order them if we were give them pell-mell. I fact, below I will order them differetly. 3. I four of the problems, the umbers k ad i the problem ames ad i the formulas appear reversed to what we are used to. Below I will switch k ad to the usual order. 4. Two of the problems are cumulatios of partitios - but there are o correspodig problems for cumulatios of compositios, combiatios or arragemets. 5. All the coutig formulas except oe (the sum) ca be expressed briefly with either the biomial coefficet ( k), the secod Stirlig umber { k}, or the partitio fuctio [ k]. 4

5 Biomial coefficiets Stirlig umbers Partitio fuctio The termiology for these 3 similar umbers is icoheret (coefficiet, fuctio, umber). Direct formulas: ( k) =! k!(k)! Recursio formulas: ( =1 1) ( k) = ( 1 k1) + ( 1 k ) Special values: >) =0 ( =1 ) ( Triagular arrays: { k} kl( = 1 k! k (l) l k 1 l ) [ k] : oe { 1} =1 { k} = { k1} 1 +k { 1 k } [ 1] =1 [ k] = [ k1] 1 + [ k k ] { } =1 { >} =0 [ ] [ >] =1 =0 (,k) (,k) (,k) The 10 coutig problems explaied I will preset the coutig problems i the followig order. O the left is the stadard ame of the items to be couted (sometimes followed by alterate ames), o the right are the (equivalece classes of) fuctios to be couted. Compositios ad Partitios of a set or a umber Read the fuctios from right to left! 1. k-compositios of a -set (K sur N) 2. k-partitios of a -set S k (K sur N) 3. k-partitios of a -set S k (K ay N) 4. k-compositios of a umber (K sur N) S 5. k-partitios of a umber S k (K sur N) S 6. k-partitios of a umber S k (K ay N) S k (Multi-)Arragemets ad (Multi-)Combiatios from a set ad k reversed - read the fuctios from left to right! 7. k-arragemets from a -set (k-permutatios) K ij N 8. k-multiarragemets from a -set (k-sequeces) K ay N 9. k-combiatios from a -set (k-selectios) S k (K ij N) 10. k-multicombiatios from a -set (k-multiselectios) S k (K ay N) 5

6 Compositios ad Partitios of a set or a umber 1. k-compositios of a -set = k-vector of disjuct subsets whose uio is the whole -set The 1-set {a} has 1 compositio: {a}. The 2-set {a,b} has 3 compositios: 1 1-compositio {a,b}; 2 2-compositios a b, b a. The 3-set {a,b,c} has 13 compositios: 1 1-compositio {a,b,c}; 6 2-compositios a {b,c}, b {a,c}, c {a,b}, {b,c} a, {a,c} b, {a,b} c; 6 3-compositios a b c, b c a, c a b, a c b, b a c, c b a. The umber of k-compositios of a -set is k! { The umber of compositios of a -set is F()= 3. k-partitios of a -set = represetatio of the -set as a uio of k-subsets The 1-set {a} has 1 partitio: {a}. The 2-set {a,b} has 2 partitios: 1 1-partitio {a,b}; 1 2-partitio a b. The 3-set {a,b,c} has 5 partitios: 1 1-partitio {a,b,c}; 3 2-partitios a {b,c}, b {a,c}, c {a,b}; 1 3-partitio a b c. The umber of k-partitios of a -set is { k} : secod Stirlig No. k} 1 k! { : Fubii-No (A000670) k} The umber of partitios of a -set is B()= 1 { : Bell-No (A000110) k} 3. k-partitios of a -set This simply meas cumulatig the previous k-partitios. The umber of k-partitios of a -set is 1 { 4. k-compositios of the umber k} = k-vectors of positive atural umbers with the sum. The umber 1 has 1 compositio: 1. The umber 2 has 2 compositios: 1 1-compositio 2; 1 2-compositio 1+1. The umber 3 has 4 compositios: 1 1-compositio 3; 6

7 2 2-compositios 2+1, 1+2; 1 3-compositio The umber 4 has 8 compositios: 1 1-compositio 4; 3 2-compositios 3+1, 1+3, 2+2; 3 3-compositios 2+1+1, 1+2+1, 1+1+2; 1 4-compositio The umber of k-compositios of the umber is ( 1 k1) The umber of compositios of the umber is 1 ( 1 5. k-partitios of the umber = k-sets of positive atural umbers with the sum. The umber 1 has 1 partitio: 1. The umber 2 has 2 partitios: 1 1-partitio 2; 1 2-partitio 1+1. The umber 3 has 3 partitios: 1 1-partitio 3; 1 2-partitio 2+1; 1 3-partitio The umber 4 has 5 partitios: 1 1-partitio 4; 2 2-partitios 3+1, 2+2; 1 3-partitio 2+1+1; 1 4-partitio The umber of k-partitios of the umber is P k ()= [ The umber of partitios of is P()= (k ) 1 [ 6. k-partitios of the umber k] = [ 2 This simply meas cumulatig the previous k-partitios. The umber of k-partitios of the umber is 1 [ k1) =21 k] ]: Partitio-fuctio (A000041) k] = [ +k k ] (Multi-)Arragemets ad (Multi-)Combiatios from a set 5. k-combiatios from a -set = k-subset of the -set The 1-set {a} has 2 combiatios: 1 0-combiatio Ø; 1 1-combiatio {a}. The 2-set {a,b} has 4 combiatios: 7

8 1 0-combiatio Ø; 2 1-combiatios {a}, {b}; 1 2-combiatio {a,b}. The 3-set {a,b,c} has 8 combiatios: 1 0-combiatio Ø; 3 1-combiatios {a}, {b}, {c}; 3 2-combiatios {b,c}, {a,c}, {a,b}; 1 3-combiatio {a,b,c}. The umber of k-combiatios from a -set is ( k) =k (" choose k") k! The umber of combiatios from a -set is 0 ( k) =2 6. k-arragemets from a -set = k-vector of distict elemets from the -set The 1-set {a} has 2 arragemets: 1 0-arragemet Ø; 1 1-arragemet a. The 2-set {a,b} has 5 arragemets: 1 0-arragemet Ø; 2 1-arragemets a, b; 2 2-arragemets (a,b), (ba). The 3-set {a,b,c} has 16 arragemets: 1 0-arragemet Ø; 3 1-arragemets a, b, c; 6 2-arragemets (b,c), (c,b), (a,c), (c,a), (a,b), (b,a); 6 3-arragemets (a,b,c), (b,c,a), (c,a,b), (a,c,b), (b,a,c), (c,b,a). The umber of k-arragemets from a -set is k = (1)... (k+1) The umber of arragemets from a -set is 0 k =! e (where x =floor(x)) 7. k-multicombiatios from a -set = k-multisubsets of the -set (the elemets of a multiset come with a multiplicity 1) The 1-set {a} has 1 0-multicombiatio Ø 1 1-multicombiatio {a}; 1 2-multicombiatio {a,a}; 1 3-multicombiatio {a,a,a};... The 2-set {a,b} has 1 0-multicombiatio Ø; 2 1-multicombiatios {a}, {b}; 3 2-multicombiatios {a,a}, {a,b}, {b,b}; 4 3-multicombiatios {a,a,a},{a,a,b},{a,b,b},{b,b,b};... 8

9 The umber of k-multicombiatios from a -set is ( +k1 k ) =k k! k is ulimited, ad the sum over k diverges. 8. k-multiarragemets from a -set = k-vectors of elemets (ot ecessarily distict) from the -set The 1-set {a} has 1 0-multiarragemet Ø; 1 1-multiarragemet a; 1 2-multiarragemet (a,a); 1 3-multiarragemet (a,a,a)... The 2-set {a,b} has 1 0-multiarragemet Ø; 2 1-multiarragemets a, b; 4 2-multiarragemets (a,a), (a,b), (b,a), (b,b); 8 3-multiarragemets (a,a,a), (a,a,b), (a,b,a), (b,a,a),... The umber of k-multiarragemets from a -set is k. k is ulimited, ad the sum over k diverges. Summig over k (a,b,b), (b,a,b), (b,b,a), (b,b,b). Here is a compilatio of the sum-over-k formulas give above. (" multichoose k") Cout OEIS # Compositios of a set F() (1) A Partitios of a set B() (1) A Compositios of a umber Partitios of a umber P() (1) A Arragemets from a set!e A Combiatios from a set I multicombiatios ad multiarragemets, k is ulimited, ad the sum over k diverges. 9