scholarly journals Lehmer code transforms and Mahonian statistics on permutations

2013 ◽  
Vol 313 (5) ◽  
pp. 581-589 ◽  
Author(s):  
Vincent Vajnovszki
Keyword(s):  
Mahonian Statistics ◽  
Lehmer Code

scholarly journals A Consecutive Lehmer Code for Parabolic Quotients of the Symmetric Group

2021 ◽  
Vol 28 (3) ◽  
Author(s):  
Wenjie Fang ◽  
Henri Mühle ◽  
Jean-Christophe Novelli
Keyword(s):  
Symmetric Group ◽  
Simple Proof ◽  
Open Problem ◽  
Lattice Isomorphism ◽  
Tamari Lattice ◽  
Lehmer Code

In this article we define an encoding for parabolic permutations that distinguishes between parabolic $231$-avoiding permutations. We prove that the componentwise order on these codes realizes the parabolic Tamari lattice, and conclude a direct and simple proof that the parabolic Tamari lattice is isomorphic to a certain $\nu$-Tamari lattice, with an explicit bijection. Furthermore, we prove that this bijection is closely related to the map $\Theta$ used when the lattice isomorphism was first proved in (Ceballos, Fang and Mühle, 2020), settling an open problem therein.

Euler–Mahonian Statistics on Ordered Set Partitions

2008 ◽  
Vol 22 (3) ◽  
pp. 1105-1137 ◽  
Author(s):  
Masao Ishikawa ◽  
Anisse Kasraoui ◽  
Jiang Zeng
Keyword(s):  
Ordered Set ◽  
Set Partitions ◽  
Ordered Set Partitions ◽  
Mahonian Statistics

scholarly journals New Euler–Mahonian Statistics on Permutations and Words

1997 ◽  
Vol 18 (3) ◽  
pp. 237-270 ◽  
Author(s):  
Robert J Clarke ◽  
Einar Steingrı́msson ◽  
Jiang Zeng
Keyword(s):  
Mahonian Statistics

scholarly journals The derangement problem relative to the Mahonian process

2003 ◽  
Vol 2003 (24) ◽  
pp. 1497-1508
Author(s):  
Jessica Delfert ◽  
Hillary Einziger ◽  
Don Rawlings
Keyword(s):  
Mahonian Statistics

We surveyn!plusq-derangement problems. Solutions to four Mahonian statistics arising in connection with cycle placement rules are presented. A few conjectures are also made.

scholarly journals Graphical Mahonian Statistics on Words

10.37236/6263 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Amy Grady ◽  
Svetlana Poznanović
Keyword(s):  
Directed Graphs ◽  
Permutation Statistics ◽  
Major Index ◽  
Mahonian Statistics

Foata and Zeilberger defined the graphical major index, $\mathrm{maj}_U$, and the graphical inversion index, $\mathrm{inv}_U$, for words over the alphabet $\{1, 2, \dots, n\}$. These statistics are a generalization of the classical permutation statistics $\mathrm{maj}$ and $\mathrm{inv}$ indexed by directed graphs $U$. They showed that $\mathrm{maj}_U$ and $\mathrm{inv}_U$ are equidistributed over all rearrangement classes if and only if $U$ is bipartitional. In this paper we strengthen their result by showing that if $\mathrm{maj}_U$ and $\mathrm{inv}_U$ are equidistributed on a single rearrangement class then $U$ is essentially bipartitional. Moreover, we define a graphical sorting index, $\mathrm{sor}_U$, which generalizes the sorting index of a permutation. We then characterize the graphs $U$ for which $\mathrm{sor}_U$ is equidistributed with $\mathrm{inv}_U$ and $\mathrm{maj}_U$ on a single rearrangement class. 

scholarly journals A New Bijective Proof of Babson and Steingrímsson's Conjecture

10.37236/6411 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Joanna N. Chen ◽  
Shouxiao Li
Keyword(s):  
Generating Functions ◽  
Permutation Patterns ◽  
Bijective Proof ◽  
Major Index ◽  
Mahonian Statistics ◽  
Pattern Functions

Babson and Steingrímsson introduced generalized permutation patterns and showed that most of the Mahonian statistics in the literature can be expressed by the combination of generalized pattern functions. Particularly, they defined a new Mahonian statistic in terms of generalized pattern functions, which is denoted $stat$. Given a permutation $\pi$, let $des(\pi)$ denote the descent number of $\pi$ and $maj(\pi)$ denote the major index of $\pi$. Babson and Steingrímsson conjectured that $(des,stat)$ and $(des,maj)$ are equidistributed on $S_n$. Foata and Zeilberger settled this conjecture using q-enumeration, generating functions and Maple packages ROTA and PERCY. Later, Burstein provided a bijective proof of a refinement of this conjecture. In this paper, we give a new bijective proof of this conjecture.

scholarly journals On the distribution of some Euler-Mahonian statistics

2015 ◽  
Vol 6 (3) ◽  
pp. 273-284
Author(s):  
Alexander Burstein
Keyword(s):  
Mahonian Statistics
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