论文标题
Mahonian和Euler-Mahonian固定统计数据
Mahonian and Euler-Mahonian statistics for set partitions
论文作者
论文摘要
集合$ [n]的分区：= \ {1,2，\ ldots，n \} $是$ [n] $的脱节非相关子集（或块）的集合，其联合为$ [n] $。在本文中，我们考虑以下很少用于设置分区的表示：给定$ [n] $的分区$ b_ {1}，b_ {2}，\ ldots，\ ldots，b_ {m} $满足$ \ max b_ max b_ max b_ {1} <\ max b_ max b_ max b_ {2} <\ max max cdots < $ w = w_ {1} w_ {2} \ ldots w_ {n} $，使得b_ {w_ {i}} $，$ i \，$ 1 \ leq i \ leq n $。我们证明，Mahonian Statistics Inv，Maj $ _ {D} $，$ R $ -MMAJ，Z，DEN，MAK，MAD，MAD，通过此表示，在设定的分区中均等分配，并且Euler-Mahonian统计数据（DES，MAJ，MAJ，MAJ），（MSTC），（MSTC，Inv），（exc，des，des），（des，des，（des，des，bak）都是均等的。
A partition of the set $[n]:=\{1,2,\ldots,n\}$ is a collection of disjoint nonempty subsets (or blocks) of $[n]$, whose union is $[n]$. In this paper we consider the following rarely used representation for set partitions: given a partition of $[n]$ with blocks $B_{1},B_{2},\ldots,B_{m}$ satisfying $\max B_{1}<\max B_{2}<\cdots<\max B_{m}$, we represent it by a word $w=w_{1}w_{2}\ldots w_{n}$ such that $i\in B_{w_{i}}$, $1\leq i\leq n$. We prove that the Mahonian statistics INV, MAJ, MAJ$_{d}$, $r$-MAJ, Z, DEN, MAK, MAD are all equidistributed on set partitions via this representation, and that the Euler-Mahonian statistics (des, MAJ), (mstc, INV), (exc, DEN), (des, MAK) are all equidistributed on set partitions via this representation.
