Enumeration of m-tuples of permutations and a new class of power bases for the space of symmetric functions

Abstract

In this paper, we continue a long line of research which shows that many generating function identities for various permutation statistics arise from well known symmetric function identities by applying certain ring homomorphisms on the ring of symmetric functions. This idea was first introduced in a 1993 paper of Brenti who used it to show that the generating functions of permutations of the symmetric group S n by descents and excedances could be derived in such a manner. In this paper, we define certain (q, t)-analogues of Brenti's homomorphism that lead to generating functions for statistics on m-tuples of permutations. Our results generalize previous work of generating functions for permutations statistics due to Carlitz, Stanley, and Fedou and Rawlings. We also introduce some new bases of symmetric functions which are necessary to extend Brenti's results on generating functions for permutations by excedances to m-tuples of permutations. Finally, we study the image of our homomorphisms on analogues of the q-basis of symmetric functions studied by Ram and King and Wybourne to describe the irreducible characters of the Hecke algebras of type A.

Permutations and words counted by consecutive patterns

“…Later, this statistic was studied by Jean-Marc Fédou with Don Rawlings and Thomas Langley with Jeff Remmel; the latter authors used the same approach of manipulating the relationships between symmetric functions we are taking [16,17,22]. This paper, however, marks the first time τ -common matches have been studied for τ other than the permutation 2 1.…”

Consecutive Up-Down Patterns in Up-Down Permutations

“…Next we define a class of symmetric functions p n,ν which have a relationship with e λ that is analogous to the relationship between h n and e λ . These functions were first introduced in [17] and [21]. Let ν be a function which maps the set of non-negative integers into the field F .…”

A reciprocity method for computing generating functions over the set of permutations with no consecutive occurrence of a permutation pattern

“…Let c-τ -mch(σ) be the number of cycle-τ -matches in the permutation σ. For example, if τ = 2 1 3 and σ = (4, 7, 5, 8, 6)(2, 3) (1,10,9), then 9 1 10 is a cycle-τ -match in the third cycle and 7 5 8 and 6 4 7 are cycle-τ -matches in the first cycle so that c-τ -mch(σ) = 3.…”

“…Letσ be the permutation that arises from C 1 · · · C k by erasing all the parentheses and commas. For example, if σ = (7,10,9,11) (4,8,6) (1,5,3,2), then σ = 7 10 9 11 4 8 6 1 5 3 2. It is easy to see that the minimal elements of the cycles correspond to left-to-right minima inσ.…”

“…That is, Remmel and various coauthors [1,9,10,11,12,13,14] developed a method called the homomorphism method to show that many generating functions involving permutation statistics can be derived by applying a homomorphism defined on the ring of symmetric functions Λ in infinitely many variables x 1 , x 2 , . .…”