Colored Posets and Colored Quasisymmetric Functions

Hsiao, Samuel K.; Petersen, T. Kyle

doi:10.1007/s00026-010-0059-0

Cited by 13 publications

(10 citation statements)

References 27 publications

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“…For all partitions λ, µ s λ (x)s µ (y) = Q∈SYT(λ,µ) F sDes(Q) (x, y).Proof. This statement follows from[25, Corollary 8] and Proposition 2.6. For a direct proof, it suffices to describe a bijection from the set of all pairs of semistandard Young tableaux S and T of shape λ and µ, respectively, to the set of all pairs (Q, u) of standard Young bitableau Q ∈ SYT(λ, µ) and monomials u which appear in the expansion of F sDes(Q) (x, y), such that if (Q, u) corresponds to (S, T ) then x S y T = u.…”

mentioning

confidence: 68%

“…Different type B analogues of quasisymmetric functions have been suggested [14,30]. The B n -analogues of the fundamental quasisymmetric functions that we need were introduced (in the more general setting of r-colored permutations) by Poirier [32,Section 3] and were further studied in [10,25]. For a signed set σ = (S, ε) ∈ Σ B (n) define…”

Section: Quasisymmetric Functionsmentioning

confidence: 99%

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Character formulas and descents for the hyperoctahedral group

Adin

Athanasiadis

Elizalde

et al. 2017

Advances in Applied Mathematics

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A general setting to study a certain type of formulas, expressing characters of the symmetric group S n explicitly in terms of descent sets of combinatorial objects, has been developed by two of the authors. This theory is further investigated in this paper and extended to the hyperoctahedral group B n . Key ingredients are a new formula for the irreducible characters of B n , the signed quasisymmetric functions introduced by Poirier, and a new family of matrices of Walsh-Hadamard type. Applications include formulas for natural B n -actions on coinvariant and exterior algebras and on the top homology of a certain poset in terms of the combinatorics of various classes of signed permutations, as well as a B n -analogue of an equidistribution theorem of Désarménien and Wachs.

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mentioning

confidence: 68%

Section: Quasisymmetric Functionsmentioning

confidence: 99%

Character formulas and descents for the hyperoctahedral group

Adin

Athanasiadis

Elizalde

et al. 2017

Advances in Applied Mathematics

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“…(This does not sound like much of a generalization when stated like this, but as we have seen the behavior of the power series Γ Z (P, γ) depends strongly on what Z is, and is not all anticipated by the Z = N × {+, −} case.) A different generalization of enriched (P, γ)partitions (introduced by Hsiao and Petersen in [HsiPet10]) are the colored (P, γ)-partitions, where the two-element set {+, −} is replaced by the set 1, ω, . .…”

Section: Exterior Peaksmentioning

confidence: 99%

Shuffle-Compatible Permutation Statistics II: The Exterior Peak Set

Grinberg

2018

Electron. J. Combin.

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This paper is a continuation of the work "Shuffle-compatible permutation statistics" by Gessel and Zhuang (but can be read independently from the latter). We study the shuffle-compatibility of permutation statistics — a concept introduced by Gessel and Zhuang, although various instances of it have appeared throughout the literature before. We prove that (as Gessel and Zhuang have conjectured) the exterior peak set statistic (Epk) is shuffle-compatible. We furthermore introduce the concept of an "LR-shuffle-compatible" statistic, which is stronger than shuffle-compatibility. We prove that Epk and a few other statistics are LR-shuffle-compatible. Furthermore, we connect these concepts with the quasisymmetric functions, in particular the dendriform structure on them.

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“…This mapping is implicit in Stanley's work (see also [10]). The fact that one has a sub-bialgebra and a homomorphism is already due to [5] (see also [18] and [11]). We give further properties of this homomorphism: it is an isometry, and preserves the internal product (this product extends the product of permutations; it is defined in Section 2.3).…”

Section: Introductionmentioning

confidence: 96%

A self paired Hopf algebra on double posets and a Littlewood–Richardson rule

Malvenuto

Reutenauer

2011

Journal of Combinatorial Theory, Series A

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Let D be the set of isomorphism types of finite double partially ordered sets, that is sets endowed with two partial orders. On ZD we define a product and a coproduct, together with an internal product, that is, degree-preserving. With these operations ZD is a Hopf algebra. We define a symmetric bilinear form on this Hopf algebra: it counts the number of pictures (in the sense of Zelevinsky) between two double posets. This form is a Hopf pairing, which means that product and coproduct are adjoint each to another. The product and coproduct correspond respectively to disjoint union of posets and to a natural decomposition of a poset into order ideals. Restricting to special double posets (meaning that the second order is total), we obtain a notion equivalent to Stanley's labelled posets, and a Hopf subalgebra already considered by Blessenohl and Schocker. The mapping which maps each double poset onto the sum of the linear extensions of its first order, identified via its second (total) order with permutations, is a Hopf algebra homomorphism, which is isometric and preserves the internal product, onto the Hopf algebra of permutations, previously considered by the two authors. Finally, the scalar product between any special double poset and double posets naturally associated to integer partitions is described by an extension of the Littlewood-Richardson rule.

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Colored Posets and Colored Quasisymmetric Functions

Cited by 13 publications

References 27 publications

Character formulas and descents for the hyperoctahedral group

Character formulas and descents for the hyperoctahedral group

Shuffle-Compatible Permutation Statistics II: The Exterior Peak Set

A self paired Hopf algebra on double posets and a Littlewood–Richardson rule

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