Two enumerative results on cycles of permutations

Answering a question of Bona, it is shown that for n≥2 the probability that 1 and 2 are in the same cycle of a product of two n-cycles on the set {1,2,…,n} is 1/2 if n is odd and 1/2 - 2/(n-1)(n+2) if n is even. Another result concerns the polynomial P[subscript λ](q) = ∑[subscript w]q[superscript κ]((1,2,…,n)⋅w), where w ranges over all permutations in the symmetric group S[subscript n] of cycle type λ, (1,2,…,n) denotes the n-cycle 1→2→⋯→n→1, and κ(v) denotes the number of cycles of the permutation v. A formula is obtained for P[subscript λ](q) from which it is deduced that all zeros of P[subscript λ](q) have real part 0.National Science Foundation (U.S.) (Grant 0604423

Long Cycle Factorizations: Bijective Computation in the General Case

This paper is devoted to the computation of the number of ordered factorizations of a long cycle in the symmetric group where the number of factors is arbitrary and the cycle structure of the factors is given. Jackson (1988) derived the first closed form expression for the generating series of these numbers using the theory of the irreducible characters of the symmetric group. Thanks to a direct bijection we compute a similar formula and provide the first purely combinatorial evaluation of these generating series

Bijective Enumeration of Bicolored Maps of Given Vertex Degree Distribution

We derive a new formula for the number of factorizations of a full cycle into an ordered product of two permutations of given cycle types. For the first time, a purely combinatorial argument involving a bijective description of bicolored maps of specified vertex degree distribution is used. All the previous results in the field rely either partially or totally on a character theoretic approach. The combinatorial proof relies on a new bijection extending the one in [G. Schaeffer and E. Vassilieva. $\textit{J. Comb. Theory Ser. A}$, 115(6):903―924, 2008] that focused only on the number of cycles. As a salient ingredient, we introduce the notion of thorn trees of given vertex degree distribution which are recursive planar objects allowing simple description of maps of arbitrary genus. \pa

Enumeration of Factorizations in the Symmetric Group: From Centrality to Non-centrality

The character theory of the symmetric group is a powerful method of studying enu- merative questions about factorizations of permutations, which arise in areas including topology, geometry, and mathematical physics. This method relies on having an encoding of the enumerative problem in the centre Z(n) of the algebra C[S_n] spanned by the symmetric group S_n. This thesis develops methods to deal with permutation factorization problems which cannot be encoded in Z(n). The (p,q,n)-dipole problem, which arises in the study of connections between string theory and Yang-Mills theory, is the chief problem motivating this research. This thesis introduces a refinement of the (p,q,n)-dipole problem, namely, the (a,b,c,d)- dipole problem. A Join-Cut analysis of the (a,b,c,d)-dipole problem leads to two partial differential equations which determine the generating series for the problem. The first equation determines the series for (a,b,0,0)-dipoles, which is the initial condition for the second equation, which gives the series for (a,b,c,d)-dipoles. An analysis of these equa- tions leads to a process, recursive in genus, for solving the (a,b,c,d)-dipole problem for a surface of genus g. These solutions are expressed in terms of a natural family of functions which are well-understood as sums indexed by compositions of a binary string. The combinatorial analysis of the (a,b,0,0)-dipole problem reveals an unexpected fact about a special case of the (p,q,n)-dipole problem. When q=n−1, the problem may be encoded in the centralizer Z_1(n) of C[S_n] with respect to the subgroup S_{n−1}. The algebra Z_1(n) has many combinatorially important similarities to Z(n) which may be used to find an explicit expression for the genus polynomials for the (p,n−1,n)-dipole problem for all values of p and n, giving a solution to this case for all orientable surfaces. Moreover, the algebraic techniques developed to solve this problem provide an alge- braic approach to solving a class of non-central problems which includes problems such as the non-transitive star factorization problem and the problem of enumerating Z_1- decompositions of a full cycle, and raise intriguing questions about the combinatorial significance of centralizers with respect to subgroups other than S_{n−1}

On the Number of Factorizations of a Full Cycle

Abstract. We give a new expression for the number of factorizations of a full cycle into an ordered product of permutations of specified cycle types. This is done through purely algebraic means, extending recent work of Biane [Nombre de factorisations d’un grand cycle, Sém. Lothar. de Combinatoire 51 (2004)]. We deduce from our result a remarkable formula of Poulalhon and Schaeffer [Factorizations of large cycles in the symmetric group, Discrete Math. 254 (2002), 433–458] that was previously derived through an intricate combinatorial argument. Résumé. Nous proposons une nouvelle formule pour le nombre de factorisations d’un grand cycle en un produit ordonné de permutations de types cycliques donnés. Nous utilisons des arguments purement algébriques, étendant un travail récent de Biane [Nombre de factorisations d’un grand cycle., Sém. Lothar. de Combinatoire 51 (2004)]. Nous déduisons de notre résultat une formule remarquable de Poulalhon et Schaeffer [Factorizations of large cycles in the symmetric group, Discrete Math. 254 (2002), 433–458] obtenue précédemment à l’aide d’arguments combinatoires complexes. 1. Notation Our notation is generally consistent with Macdonald [5]. We write λ ⊢ n (or |λ | = n) and ℓ(λ) = k to indicate that λ is a partition of n into k parts; that is, λ = (λ1,..., λk) with λ1 ≥ · · · ≥ λk ≥ 1 and λ1+...+λk = n. If λ has exactly mi parts equal to i then we write λ = [1m12m2 · · ·], suppressing terms with mi = 0. We also define zλ = � i imimi! and Aut(λ) = � i mi!. A hook is a partition of the form [1b, a + 1] with a, b ≥ 0. We use Frobenius notation for hooks, writing (a|b) in place of [1b, a + 1]. The conjugacy class of the symmetric group Sn consisting of all n!/zλ permutations of cycle type λ ⊢ n will be denoted by Cλ. The irreducible characters χλ of Sn are naturally indexed by partitions λ of n, and we use the usual notation χλ µ for the common value of χλ at any element of Cµ. We write fλ for the degree χλ [1n] of χλ. For vectors j = (j1,...,jm) and x = (x1,..., xm) we use the abbreviations j! = j1! · · ·jm! and xj = x j1 1 · · · xjm m. Finally, if α ∈ Q and f ∈ Q[[x]] is a formal power series, then we write [αxj] f(x) for the coefficient of the monomial αxj in f(x)