Inversion-like and Major-like Statistics of an Ordered Partition of a Multiset
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Given a partition $\ld=(\ld_1, \ld_2, \ldots, \ld_l)$ of a positive integer $n$,let ${\rm Tab}(\ld,k)$ be the set of all tabloids of shape $\ld$ whose weightsrange over the set of all $k$-compositions of $n$and $\mathcal{OP}_{\ld^{\rm rev}}^k$ the set of all ordered partitions into $k$ blocksof the multiset $\{1^{\ld_l}2^{\ld_{l-1}}\cdots l^{\ld_1}\}$. In \cite{B}, Butler introduced an inversion-like statistic on ${\rm Tab}(\ld,k)$to show that the rank-selected M\"{o}bius invariantarising from the subgroup lattice of a finite abelian $p$-group of type $\ld$has nonnegative coefficients as apolynomial in $p$. In this paper, we introduce an inversion-like statistic on the set of orderedpartitions of a multiset and construct an inversion-preserving bijection between${\rm Tab}(\ld,k)$ and $\mathcal{OP}_{\hat \ld}^k$. When $k=2$, we also introduce a major-like statistic on ${\rm Tab}(\ld,2)$and study its connection to the inversion statistic due to Butler.
Abstract
Given a partition $\ld=(\ld_1, \ld_2, \ldots, \ld_l)$ of a positive integer $n$,let ${\rm Tab}(\ld,k)$ be the set of all tabloids of shape $\ld$ whose weightsrange over the set of all $k$-compositions of $n$and $\mathcal{OP}_{\ld^{\rm rev}}^k$ the set of all ordered partitions into $k$ blocksof the multiset $\{1^{\ld_l}2^{\ld_{l-1}}\cdots l^{\ld_1}\}$. In \cite{B}, Butler introduced an inversion-like statistic on ${\rm Tab}(\ld,k)$to show that the rank-selected M\"{o}bius invariantarising from the subgroup lattice of a finite abelian $p$-group of type $\ld$has nonnegative coefficients as apolynomial in $p$. In this paper, we introduce an inversion-like statistic on the set of orderedpartitions of a multiset and construct an inversion-preserving bijection between${\rm Tab}(\ld,k)$ and $\mathcal{OP}_{\hat \ld}^k$. When $k=2$, we also introduce a major-like statistic on ${\rm Tab}(\ld,2)$and study its connection to the inversion statistic due to Butler.
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