Several variants of the Dumont differential system and permutation statistics

Abstract

The Dumont differential system on the Jacobi elliptic functions was introduced by Dumont (Math Comp, 1979, 33: 1293-1297 and was extensively studied by Dumont, Viennot, Flajolet and so on. In this paper, we first present a labeling scheme for the cycle structure of permutations. We then introduce two types of Jacobi-pairs of differential equations. We present a general method to derive the solutions of these differential equations. As applications, we present some characterizations for several permutation statistics.

A symmetry on weakly increasing trees and multiset Schett polynomials

“…As another application of Theorems 3.3 and 3.1, we can confirm affirmatively a conjecture posed by Ma-Mansour-Wang-Yeh [18]. Actually, the context-free grammar G in (3.4) was first considered in [18], as a conjunction of the two grammars in (3.1) and (3.2). Define the integer sequences t n,i,j by…”

“…One is to construct the involution r p q : T M Ñ T M to prove combinatorially the refined symmetry (1.2), which is provided in Section 2, as well as several relevant interesting consequences. The other is to study the algebraic aspect of the refined symmetry (1.2), including proofs of Theorem 1.6 and a conjecture of Ma-Mansour-Wang-Yeh [18] via Chen's context-free grammar and a generating function proof of (1.2), which are fulfilled in Section 3.…”

“…We will show Theorem 1.6 by employing Chen's context-free grammar. As an application, a conjecture posed at the end of [18] is solved. In addition, we manage to extend a regular generating function proof of (1.2) for plane trees to weakly increasing trees.…”

“…Increasing trees and Jacobi elliptic functions. The context-free grammar was introduced by Chen in [3] and has been found useful in studying various combinatorial structures [3,4,11,18], including permutations, increasing trees, labeled rooted trees and set partitions.…”

Some multivariate polynomials for doubled permutations

“…These functions appear in a variety of problems in physics and have been extensively studied in mathematical physics, algebraic geometry, combinatorics and number theory (see [5,6,8,9,11,12,19,20,21,27,28] for instance). When x = 0 or x = 1, the Jacobi elliptic functions degenerate into trigonometric or hyperbolic functions: sn (z, 0) = sin z, cn (z, 0) = cos z, dn (z, 0) = 1, sn (z, 1) = tanh z, cn (z, 1) = dn (z, 1) = sech z.…”