Two Enumerative Results on Cycles of Permutations 1

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1 Two Enumeratve Results on Cycles of Permutatons Rchard P. Stanley Department of Mathematcs Massachusetts Insttute of Technology Cambrdge, MA 039, USA In memory of Tom Brylawsk verson of 5 Aprl 009 Abstract Answerng a queston of Bóna, t s shown that for n the probablty that and are n the same cycle of a product of two n-cycles on the set {,,..., n} s / f n s odd and (n )(n+) f n s even. Another result concerns the polynomal P λ (q) = w qκ((,,...,n) w), where w ranges over all permutatons n the symmetrc group S n of cycle type λ, (,,..., n) denotes the n-cycle n, and κ(v) denotes the number of cycles of the permutaton v. A formula s obtaned for P λ (q) from whch t s deduced that all zeros of P λ (q) have real part 0. Introducton. Let λ = (λ, λ,... ) be a partton of n, denoted λ n. In general, we use notaton and termnology nvolvng parttons and symmetrc functons from [, Ch. 7]. Let S n denote the symmetrc group of all permutatons of [n] = {,,..., n}. If w S n then wrte ρ(w) = λ f w has cycle type λ,.e., f the (nonzero) λ s are the lengths of the cycles of w. The conjugacy classes of S n are gven by K λ = {w S n : ρ(w) = λ}. The class multplcaton problem for S n may be stated as follows. Gven λ, µ, ν n, how many pars (u, v) S n S n satsfy u K λ, v K µ, Ths materal s based upon work supported by the Natonal Scence Foundaton under Grant No Any opnons, fndngs and conclusons or recommendatons expressed n ths materal are those of the author and do not necessarly reflect those of the Natonal Scence Foundaton.

2 uv K ν? The case when one of the parttons s (n) (.e., one of the classes conssts of the n-cycles) s partcularly nterestng and has receved much attenton. For a sample of some recent work, see [][6] [9]. In ths paper we make two contrbutons to ths subject. For the frst, we solve a problem of Bóna and Flynn [4] askng what s the probablty that two fxed elements of [n] le n the same cycle of the product of two random n-cycles. In partcular, we prove the conjecture of Bóna that ths probablty s / when n s odd. Our method of proof s an ugly computaton based on a formula of Boccara []. The technque can be generalzed, and as an example we compute the probablty that three fxed elements of [n] le n the same cycle of the product of two random n-cycles. For our second result, let κ(w) denote the number of cycles of w S n, and let (,,..., n) denote the n-cycle n. For λ n, defne the polynomal P λ (q) = q κ((,,...,n) w). () ρ(w)=λ In Theorem 3. we obtan a formula for P λ (q). We also prove from ths formula (Corollary 3.3) that every zero of P λ (q) has real part 0. A problem of Bóna. Let π n denote the probablty that f two n-cycles u, v are chosen unformly at random n S n, then and (or any two elements and j by symmetry) appear n the same cycle of the product uv. Mklós Bóna conjectured (prvate communcaton) that π n = / f n s odd, and asked about the value when n s even. For the reason behnd ths conjecture, see Bóna and Flynn [4]. In ths secton we solve ths problem. Let us note that t s easy to see (a straghtforward generalzaton of [3, Prop. 6.8]) that the probablty that,,..., k appear n the same cycle of a random permutaton n S n s /k for k n. Theorem.. For n we have { π n =, n odd, n even. (n )(n+)

3 Proof. Frst note that f w S n has cycle type λ, then the probablty that and are n the same cycle of w s ( λ ) λ (λ q λ = ( ) n =. ) n(n ) Let a λ be the number of pars (u, v) of n-cycles n S n for whch uv has type λ. Then π n = a (n )! λ q λ. By Boccara [] the number of ways to wrte a fxed permutaton w S n of type λ as a product of two n-cycles s (n )! 0 λ n x λ (x ) λ dx. Let n!/z λ denote the number of permutatons w S n of type λ. We get n! λ (λ ) π n = (n )! n(n ) = n (n )! λ n z λ λ n z λ 0 x λ (x ) λ dx λ (λ ) 0 x λ (x ) λ dx. Now let p λ (a, b) denote the power sum symmetrc functon p λ n the two varables a, b, and let l(λ) denote the length (number of parts) of λ. It s easy to check that l(λ)+ a p λ (a, b) a b a=b= = λ (λ ). By the exponental formula (permutaton verson) [, Cor. 5..9] or by [, Prop ], ( ) z λ l(λ) p λ (a, b) x λ (x ) λ t n n 0 λ n 3

4 = exp ( a k + b k k k It follows that (n )π n s the coeffcent of t n n F (t) := [ 0 a exp a b k k ( a k + b k ) (x k (x ) k )t k. ) (x k (x ) k )t k ] a=b= We can easly perform ths computaton wth Maple, gvng F (t) = 0 t ( x tx + tx ) ( t(x ))( tx) 3 dx = t log( t ) t ( t). Extract the coeffcent of t n and dvde by n to obtan π n as clamed. It s clear that the argument used to prove Theorem can be generalzed. For nstance, usng the fact that 3 l(λ)+ 3 a a b + 3 p λ (a, b, c) a b c a=b=c= we can obtan the followng result. = λ (λ )(λ ), Theorem.. Let π n (3) denote the probablty that f two n-cycles u, v are chosen unformly at random n S n, then,, and 3 appear n the same cycle of the product uv. Then for n 3 we have dx. π (3), (n )(n+3) 3, 3 (n )(n+) n = { 3 + n odd n even. Are there smpler proofs of Theorems. and., especally Theorem. when n s odd? 4

5 3 A polynomal wth purely magnary zeros Gven λ n, let P λ (q) be defned by equaton (). Let (a) n denote the fallng factoral a(a ) (a n + ). Let E be the backward shft operator on polynomals n q,.e., Ef(q) = f(q ). Theorem 3.. Suppose that λ has length l. Defne the polynomal Then g λ (t) = t l ( t λ j ). j= P λ (q) = z λ g λ(e)(q + n ) n. () Proof. Let x = (x, x,... ), y = (y, y,... ), and z = (z, z,... ) be three dsjont sets of varables. Let H µ denote the product of the hook lengths of the partton µ (defned e.g. n [, p. 373]). Wrte s λ and p λ for the Schur functon and power sum symmetrc functon ndexed by λ. The followng dentty s the case k = 3 of [5, Prop..] and [, Exer. 7.70]: H µ s µ (x)s µ (y)s µ (z) = p ρ(u) (x)p ρ(v) (y)p ρ(w) (z). (3) n! uvw= n S n µ n For a symmetrc functon f(x) let f( q ) = f(,,...,, 0, 0,... ) (q s). Thus p ρ(w) ( q ) = q κ(w). Let χ λ (µ) denote the rreducble character of S n ndexed by λ evaluated at a permutaton of cycle type µ [, 7.8]. Recall [, Cor and Thm ] that s µ = ν n z ν χµ (ν)p ν, where #K ν = n!/z ν as above. Take the coeffcent of p n (x)p λ (y) n equaton (3) and set z = q. Snce there are (n )! n-cycles u, the rght-hand sde becomes n P λ(q). Hence P λ (q) = n µ n H µ zn χ µ (n)z λ χµ (λ)s µ ( q ). (4) Wrte σ() = n,, the hook wth one part equal to n and parts equal to, for 0 n. Now z n = n, and e.g. by [, Exer. 7.67(a)] we 5

6 have χ µ (n) = { ( ), f µ = σ(), 0 n 0, otherwse. Moreover, s σ() ( q ) = (q + n ) n H σ() by the hook-content formula [, Cor. 7..4]. Therefore we get from equaton (4) that P λ (q) = z λ n ( ) χ σ() (λ)(q + n ) n. (5) =0 The followng dentty s a smple consequence of Per s rule [, Thm ] and appears n [7, I.3, Ex. 4]: + tx n = + (t + u) s σ() t u n. ux Substtute t for t, set u = and take the scalar product wth p λ. Snce s µ, p λ = χ µ (λ) the rght-hand sde becomes ( t) n =0 ( ) χ σ() (λ)t. On the other hand, the left-hand sde s gven by ( p n exp exp ) p n p n n n tn, p λ = exp n ( tn ), p λ n n n l ( = t λ ), by standard propertes of power sum symmetrc functons [, 7.7]. Hence =0 = n ( ) χ σ() (λ)t = g λ (t). =0 Comparng wth equaton (5) completes the proof. Note.. Snce ( E)(q + n) n+ = (n + )(q + n ) n, equaton () can be rewrtten as P λ (q) = g λ (n + )z (E)(q + n) n+, (6) λ where g λ (t) = l j= ( tλ j ). 6

7 . A dfferent knd of generatng functon for the coeffcents of P λ (q) (though of course equvalent to Theorem 3.) was obtaned by D. Zager [3, Thm. ]. The zeros of the polynomal P λ (q) have an nterestng property that wll follow from the followng result. Theorem 3.. Let g(t) be a complex polynomal of degree exactly d, such that every zero of g(t) les on the crcle z =. Suppose that the multplcty of as a root of g(t) s m 0. Let P (q) = g(e)(q + n ) n. (a) If d n, then P (q) = (q + n d ) n d Q(q), where Q(q) s a polynomal of degree d m for whch every zero has real part (d n + )/. (b) If d n, then P (q) s a polynomal of degree n m for whch every zero has real part (d n + )/. Proof. Frst, the statements about the degrees of Q(q) and P (q) are clear; for we can wrte g(t) = c u (t u) and apply the factors t u consecutvely. If h(q) s any polynomal and u then deg (E u)h(q) = deg h(q), whle deg (E )h(q) = deg h(q). The remander of the proof s by nducton on d. The base case d = 0 s clear. Assume the statement for d < n. Thus for deg g(t) = d we have g(e)(q + n ) n = (q + n d ) n d Q(q) ( = (q + n d ) n d q d n + ) δ j for certan real numbers δ j. Now j (E u)g(e)(q + n ) n = (q + n d ) n d Q(q) u(q + n d ) n d Q(q ) = (q + n d ) n d [(q + n d )Q(q) u(q ) Q(q )] = (q + n d ) n d Q (q), 7

8 say. The proof now follows from a standard argument (e.g., [8, Lemma 9.3]), whch we gve for the sake of completeness. Let Q (α + β) = 0, where α, β R. Thus (α + β + n d ) ( α + β d n + ) δ j j = u(α + β ) j ( α + β d n + ) δ j. Lettng u = and takng the square modulus gves (α + n d ) + β (α ) + β If α < (d n + )/ then j α d n+ + (β δj ) α d n+ + (β δj ) =. (α + n d ) (α ) < 0 and ( α d n + ) ( < α d n + ). The nequaltes are reversed f α > (d n + )/. Hence α = (d n + )/, so the theorem s true for d n. For d n we contnue the nducton, the base case now beng d = n whch was proved above. The nducton step s completely analogous to the case d n above, so the proof s complete. Corollary 3.3. The polynomal P λ (q) has degree n l(λ)+, and every zero of P λ (q) has real part 0. Proof. The proof s mmedate from Theorem 3. and the specal case g(t) = g λ (t) (as defned n Theorem 3.) and d = n of Theorem 3.. It s easy to see from Corollary 3.3 (or from consderatons of party) that P λ (q) = ( ) n P λ ( q). Thus we can wrte { Rλ (q ), n even P λ (q) = qr λ (q ), n odd, 8

9 for some polynomal R λ (q). It follows from Corollary 3.3 that R λ (q) has (nonpostve) real zeros. In partcular (e.g., [, Thm. ]) the coeffcents of R λ (q) are log-concave wth no external zeros, and hence unmodal. The case λ = (n) s especally nterestng. Wrte P n (q) for P (n) (q). From equaton (6) we have Now and P n (q) = n(n + ) ((q + n) n+ (q) n+ ). (q) n+ = ( ) n+ ( q + n) n+ n+ (q + n) n+ = c(n +, k)q k, k= where c(n +, k) s the sgnless Strlng number of the frst knd (the number of permutatons w S n+ wth k cycles) [0, Prop..3.4]. Hence n(n + ) ((q + n) n+ (q) n+ ) = ( n+ ) k n (mod ) c(n +, k)x k. We therefore get the followng result, frst obtaned by Zager [3, Applcaton 3]. Corollary 3.4. The number of n-cycles w S n for whch w (,,..., n) has exactly k cycles s 0 f n k s odd, and s otherwse equal to c(n+, k)/ n+. Is there a smple bjectve proof of Corollary 3.4? Let λ, µ n. A natural generalzaton of P λ (q) s the polynomal P λ,µ (q) = q κ(wµ w), ρ(w)=λ where w µ s a fxed permutaton n the conjugacy class K µ. Let us pont out that t s false n general that every zero of P λ,µ (q) has real part 0. For nstance, P 33,33 (q) = q q q q, four of whose zeros are approxmately ±.366 ±

10 References [] P. Bane, Nombre de factorsatons d un grand cycle, Sém. Lothar. de Combnatore 5 (004). [] G. Boccara, Nombres de représentatons d une permutaton comme produt de deux cycles de longuers données, Dscrete Math. 9 (980) [3] M. Bóna, A Walk Through Combnatorcs, second ed., World Scentfc, Sngapore, 006. [4] M. Bóna and R. Flynn, The average number of block nterchanges needed to sort a permutaton and a recent result of Stanley, preprnt; arxv: [5] P. J. Hanlon, R. Stanley, and J. R. Stembrdge, Some combnatoral aspects of the spectra of normally dstrbuted random matrces, Contemporary Mathematcs 58 (99), [6] J. Irvng, On the number of factorzatons of a full cycle, J. Combnatoral Theory, Ser. A 3 (006), [7] I. G. Macdonald, Symmetrc Functons and Hall Polynomals, second ed., Oxford Unversty Press, Oxford, 995. [8] A. Postnkov and R. Stanley, Deformatons of Coxeter hyperplane arrangements, J. Combnatoral Theory (A) 9 (000), [9] D. Poulalhon and G. Schaeffer, Factorzatons of large cycles n the symmetrc group, Dscrete Math. 54 (00), [0] R. Stanley, Enumeratve Combnatorcs, vol., Wadsworth and Brooks/Cole, Pacfc Grove, CA, 986, x pages; second prntng, Cambrdge Unversty Press, New York/Cambrdge, 996. [] R. Stanley, Unmodal and log-concave sequences n algebra, combnatorcs, and geometry, n Graph Theory and Its Applcatons: East and West, Ann. New York Acad. Sc., vol. 576, 989, pp [] R. Stanley, Enumeratve Combnatorcs, vol., Cambrdge Unversty Press, New York/Cambrdge,

11 [3] D. Zager, On the dstrbuton of the number of cycles of elements n symmetrc groups, Neuw Arch. Wsk. 3(3) (995),