Sorting index and Mahonian–Stirling pairs for labeled forests

Abstract

Björner 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. Specifically, we introduce the notions of bottom-totop maxima, cyclic bottom-to-top maxima, sorting index, and cycle minima. Then we show that the pairs (inv, Bt-max), (sor, Cyc), and (maj, Cbt-max) are equidistributed. Our results extend the result of Björner and Wachs and generalize results for permutations. We also introduce analogous statistics for signed labeled forests and show equidistribution results which generalize results for signed permutations.

Graphical Mahonian Statistics on Words

“…The sorting index has been extended to labeled forests by the authors [8]. It can also be naturally extended to words w ∈ R(α) by a generalization of Straight Selection Sort which reorders the letters into a weakly increasing sequence.…”

“…The algorithm first places n in the n-th position by applying a transposition, then places n − 1 in the (n − 1)-st position by applying a transposition, etc. For example, for σ = 2413576, we have 2413576 The sorting index has been extended to labeled forests by the authors [8]. It can also be naturally extended to words w ∈ R(α) by a generalization of Straight Selection Sort which reorders the letters into a weakly increasing sequence.…”

$k$-Factorizations of the full cycle and generalized Mahonian statistics on $k$-forests

“…This was first proved inductively by Björner and Wachs [BW89], and later bijectively by Liang and Wachs [LW92]. More recently, Grady and Poznanović [GP16] established this result by mapping both inv(F ) and maj(F ) to a common code called a subexcedant sequence on F . We will not go into further detail here.…”