Related papers: The inversion number and the major index are asymp…
The Mahonian statistic is the number of inversions in a permutation of a multiset with $a_i$ elements of type $i$, $1\le i\le m$. The counting function for this statistic is the $q$ analog of the multinomial coefficient…
We use recurrences (alias difference equations) to prove the longstanding conjecture that the two most important permutation statistics, namely the number of inversions and the major index, are asymptotically joint-independently-normal. We…
Two well known mahonian statistics on words are the inversion number and the major index. In 1996, Foata and Zeilberger introduced generalizations, parameterized by relations, of these statistics. In this paper, we study the statistics…
Foata and Zeilberger defined the graphical major index, $\mathrm{maj}'_U$, and the graphical inversion index, $\mathrm{inv}'_U$, for words. These statistics are a generalization of the classical permutation statistics $\mathrm{maj}$ and…
Our first main result shows that, for words with a fixed multiset of weak right-to-left minima, the statistics within each of the following three classes are equidistributed: 1. Mahonian statistics: $\textsf{inv}$, $\textsf{maj}$,…
A generalization of the classical statistics ``maj'' and ``inv'' (the major index and number of inversions) on words is introduced, parameterized by arbitrary graphs on the underlying alphabet. The question of characterizing those graphs…
In 2000, Babson and Steingr\'{i}msson generalized the notion of permutation patterns to the so-called vincular patterns, and they showed that many Mahonian statistics can be expressed as sums of vincular pattern occurrence statistics. STAT…
Chatteerjee and Diaconis have recently shown the asymptotic normality for the joint distribution of the number of descents and inverse descents in a random permutation. A noteworthy point of their results is that the asymptotic variance of…
Inspired by the $k$-inversion statistic for LLT polynomials, we define a $k$-inversion number and $k$-descent set for words. Using these, we define a new statistic on words, called the $k$-major index, that interpolates between the major…
Recently Cheng et al. (Adv. in Appl. Math. 143 (2023) 102451) generalized the inversion number to partial permutations, which are also known as Laguerre digraphs, and asked for a suitable analogue of MacMahon's major index. We provide such…
For a permutation $\pi$ the major index of $\pi$ is the sum of all indices $i$ such that $\pi_i > \pi_{i+1}$. It is well known that the major index is equidistributed with the number of inversions over all permutations of length $n$. In…
We give a direct combinatorial proof of the equidistribution of two pairs of permutation statistics, (des, aid) and (lec, inv), which have been previously shown to have the same joint distribution as (exc, maj), the major index and the…
Babson and Steingr\'{\i}msson 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…
The study of Mahonian statistics dated back to 1915 when MacMahon showed that the major index and the inverse number have the same distribution on a set of permutations with length n. Since then, many Mahonian statistics have been…
Two well-known distributions in the study of permutation statistics are the Mahonian and Eulerian distributions. Mahonian statistics include the major index MAJ and the number of inversions INV, while examples of Eulerian statistics are the…
The standard algorithm for generating a random permutation gives rise to an obvious permutation statistic $\stat$ that is readily seen to be Mahonian. We give evidence showing that it is not equal to any previously published statistic. Nor…
In 2000, Babson and Steingr\'imsson introduced the notion of what is now known as a permutation vincular pattern, and based on it they re-defined known Mahonian statistics and introduced new ones, proving or conjecturing their Mahonity.…
We present a short proof of MacMahon's classic result that the number of permutations with $k$ inversions equals the number whose major index (sum of positions at which descents occur) is $k$
Bj\"orner and Wachs defined a major index for labeled plane forests and showed that it has the same distribution as the number of inversions. We define and study the distributions of a few other natural statistics on labeled forests.…
Inversion sequences, also known as subexcedant sequences, form a fundamental class of objects in enumerative combinatorics. In this paper, we study the joint distribution of five statistics on inversion sequences. While several statistics…