Cyclic Derangements

Assaf, Sami

doi:10.37236/435

Cited by 9 publications

(6 citation statements)

References 18 publications

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“…Wreath products and the incomplete gamma function. Assaf introduced a generalization of derangements to wreath products with cyclic groups [3]. Let C r ≀ S n denote the wreath product of the cyclic group of order r with the symmetric group of degree n. This group acts on [r] × [n].…”

Section: Derangements Are a Classical Topic In Combinatoricsmentioning

confidence: 99%

Derangements and the $p$-adic incomplete gamma function

O'Desky¹,

Richman²

2020

Preprint

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We introduce a new p-adic analogue of the incomplete gamma function. We also introduce a closely related family of combinatorial sequences counting derangements and arrangements in certain wreath products.This means expressions outside of the classical range of definition can be computed relative to a given prime. For instance, for any prime p, the limit d −1 := lim n→∞ d p n −1 (in the p-adic topology)

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Section: Derangements Are a Classical Topic In Combinatoricsmentioning

confidence: 99%

Derangements and the $p$-adic incomplete gamma function

O'Desky¹,

Richman²

2020

Preprint

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“…Let D r n be the set of all colored derangements in S n,r . Faliharimalala and Zeng [25, Equation (2.7)] (see also [6,Theorem 2.1], where colored derangements are called cyclic derangements) proved the following formula…”

Section: 15)mentioning

confidence: 99%

“…Euler-Mahonian identities. Equation (4.2) is Biagioli-Zeng's identity (2.5) and Equation (4.1) appears in Assaf's work[6, Equation (13) for t = 1], where the author uses the length order.…”

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confidence: 99%

Specializations of colored quasisymmetric functions and Euler-Mahonian identities

Moustakas¹

2020

Preprint

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We propose a unified approach to prove general formulas for the joint distribution of an Eulerian and a Mahonian statistic over a set of colored permutations by specializing Poirier's colored quasisymmetric functions. We apply this method to derive formulas for Euler-Mahonian distributions on colored permutations, derangements and involutions. A number of known formulas are recovered as special cases of our results, including formulas of Biagioli-Zeng, Assaf, Haglund-Loehr-Remmel, Chow-Mansour, Biagioli-Caselli, Bagno-Biagioli, Faliharimalala-Zeng. Several new results are also obtained. For instance, a two-parameter flag major index on signed permutations is introduced and formulas for its distribution and its joint distribution with some Eulerian partners are proven.

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“…, σ n ). We adopt the following total order on the elements of (Z r × [n]) ∪ {0} (see [1,2], for example):…”

Section: 3mentioning

confidence: 99%

Stable multivariate W -Eulerian polynomials

Visontai

Williams

2013

Journal of Combinatorial Theory, Series A

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We prove a multivariate strengthening of Brenti's result that every root of the Eulerian polynomial of type B is real. Our proof combines a refinement of the descent statistic for signed permutations with the notion of real stability-a generalization of realrootedness to polynomials in multiple variables. The key is that our refined multivariate Eulerian polynomials satisfy a recurrence given by a stability-preserving linear operator.Our results extend naturally to colored permutations, and we also give stable generalizations of recent real-rootedness results due to Dilks, Petersen, and Stembridge on affine Eulerian polynomials of types A and C. Finally, although we are not able to settle Brenti's real-rootedness conjecture for Eulerian polynomials of type D, nor prove a companion conjecture of Dilks, Petersen, and Stembridge for affine Eulerian polynomials of types B and D, we indicate some methods of attack and pose some related open problems. Conjecture 1.1 (Conjecture 5.2 in [9]). For every finite Coxeter group W, the descent generating polynomial W(x) has only real roots. Dilks, Petersen, and Stembridge later extended the definition of Eulerian polynomials to include affine descents, and proposed the following companion to Brenti's conjecture. Conjecture 1.2 (Conjecture 4.1 in [12]). For every finite Weyl group W, the affine descent generating polynomial W(x) has only real roots.Again, this was not completely proved-A n (x) and C n (x) were shown to have only real roots, the exceptional cases were verified, but the real-rootedness of the affine Eulerian polynomials of types B and D remains an open problem.

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Cyclic Derangements

Cited by 9 publications

References 18 publications

Derangements and the $p$-adic incomplete gamma function

Derangements and the $p$-adic incomplete gamma function

Specializations of colored quasisymmetric functions and Euler-Mahonian identities

Stable multivariate W -Eulerian polynomials

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