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):  
2008 ◽  
Vol 22 (3) ◽  
pp. 1105-1137 ◽  
Author(s):  
Masao Ishikawa ◽  
Anisse Kasraoui ◽  
Jiang Zeng

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

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.


10.37236/6263 ◽  
2018 ◽  
Vol 25 (1) ◽  
Author(s):  
Amy Grady ◽  
Svetlana Poznanović

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. 


10.37236/6411 ◽  
2017 ◽  
Vol 24 (2) ◽  
Author(s):  
Joanna N. Chen ◽  
Shouxiao Li

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.


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

2013 ◽  
Vol 244 ◽  
pp. 925-954 ◽  
Author(s):  
Matthias Beck ◽  
Benjamin Braun

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