Euler–Mahonian statistics and descent bases for semigroup algebras

Benjamin Braun; McCabe Olsen

doi:10.1016/j.ejc.2017.11.005

Euler–Mahonian statistics and descent bases for semigroup algebras

Benjamin Braun, McCabe Olsen

Mathematics

Research output: Contribution to journal › Article › peer-review

2 Scopus citations

Abstract

We consider quotients of the unit cube semigroup algebra by particular Z_r≀S_n-invariant ideals. Using Gröbner basis methods, we show that the resulting graded quotient algebra has a basis where each element is indexed by colored permutations (π,ϵ)∈Z_r≀S_n and each element encodes the negative descent and negative major index statistics on (π,ϵ). This gives an algebraic interpretation of these statistics that was previously unknown. This basis of the Z_r≀S_n-quotients allows us to recover certain combinatorial identities involving Euler–Mahonian distributions of statistics.

Original language	English
Pages (from-to)	237-254
Number of pages	18
Journal	European Journal of Combinatorics
Volume	69
DOIs	https://doi.org/10.1016/j.ejc.2017.11.005
State	Published - Mar 2018

Bibliographical note

Funding Information:
The authors would like to thank the anonymous referees for their helpful comments and suggestions. The first author was partially supported by grant H98230-16-1-0045 from the U.S. National Security Agency . The second author was partially supported by a 2016 National Science Foundation/Japanese Society for the Promotion of Science East Asia and Pacific Summer Institutes Fellowship award NSF OISE–1613525 .

Publisher Copyright:
© 2017 Elsevier Ltd

ASJC Scopus subject areas

Discrete Mathematics and Combinatorics

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abstract = "We consider quotients of the unit cube semigroup algebra by particular Zr≀Sn-invariant ideals. Using Gr{\"o}bner basis methods, we show that the resulting graded quotient algebra has a basis where each element is indexed by colored permutations (π,ϵ)∈Zr≀Sn and each element encodes the negative descent and negative major index statistics on (π,ϵ). This gives an algebraic interpretation of these statistics that was previously unknown. This basis of the Zr≀Sn-quotients allows us to recover certain combinatorial identities involving Euler–Mahonian distributions of statistics.",

author = "Benjamin Braun and McCabe Olsen",

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N2 - We consider quotients of the unit cube semigroup algebra by particular Zr≀Sn-invariant ideals. Using Gröbner basis methods, we show that the resulting graded quotient algebra has a basis where each element is indexed by colored permutations (π,ϵ)∈Zr≀Sn and each element encodes the negative descent and negative major index statistics on (π,ϵ). This gives an algebraic interpretation of these statistics that was previously unknown. This basis of the Zr≀Sn-quotients allows us to recover certain combinatorial identities involving Euler–Mahonian distributions of statistics.

AB - We consider quotients of the unit cube semigroup algebra by particular Zr≀Sn-invariant ideals. Using Gröbner basis methods, we show that the resulting graded quotient algebra has a basis where each element is indexed by colored permutations (π,ϵ)∈Zr≀Sn and each element encodes the negative descent and negative major index statistics on (π,ϵ). This gives an algebraic interpretation of these statistics that was previously unknown. This basis of the Zr≀Sn-quotients allows us to recover certain combinatorial identities involving Euler–Mahonian distributions of statistics.

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Euler–Mahonian statistics and descent bases for semigroup algebras

Abstract

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