Abstract
We derive several multivariable generating functions for a generalized pattern-matching condition on the wreath product of the cyclic group and the symmetric group . In particular, we derive the generating functions for the number of matches that occur in elements of for any pattern of length 2 by applying appropriate homomorphisms from the ring of symmetric functions over an infinite number of variables to simple symmetric function identities. This allows us to derive several natural analogues of the distribution of rises relative to the product order on elements of . Our research leads to connections to many known objects/structures yet to be explained combinatorially.
Highlights
The goal of this paper is to study pattern-matching conditions on the the wreath symmetric product Ck ≀ group Sn
We can think of the elements Ck ≀ Sn as pairs γ = (σ, ε) where σ =
Since the elementary symmetric functions eλ and the homogeneous symmetric functions hλ are both bases for Λ, it makes sense to talk about the coefficient of the homogeneous symmetric functions when written in terms of the elementary symmetric function basis
Summary
The goal of this paper is to study pattern-matching conditions on the the wreath symmetric product Ck ≀ group Sn. Mansour [3] proved via recursion that for any (τ, u) ∈ Ck ≀ S2, the number of elements in Ck ≀ Sn which avoid exact occurrences of (τ, u) is. Substituting z0 = z1 = ⋅ ⋅ ⋅ = zk−1 = 1 into (12) in these special cases will yield the following generating functions for rises, strict rises, and weak rises in Ck ≀ Sn. Theorem 5. We shall show that if NM(Υ,A⃗),n denotes the number of (σ, w) ∈ Ck ≀ Sn such that (σ, w) has no (Υ, A⃗)-matches, the sequence (NM(Υ,A⃗),n)n≥0 appears in the OIES [16] in several special cases.
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