In this paper, we introduce the concept of weakly increasing trees on a multiset M, which is an extension of plane trees and increasing trees on the set $\{0,1,\dots ,n\}$. We define the M-Eulerian–Narayana polynomial for weakly increasing trees on a multiset M, which interpolates between the Eulerian polynomial and the Narayana polynomial. We obtain a compact product formula for the number of weakly increasing trees on a general multiset. Inspired by some remarkable equidistributions between multipermutations and s-inversion sequences, we establish two connections between our M-Eulerian–Narayana polynomials for the multiset $M=\{1^2,2^2,\dots ,n^2\}$ (resp. $M=\{1^2,2^2,\dots ,{(n-1)}^2,n\}$) and Savage and Schuster's s-Eulerian polynomials for the sequence $\mathbf{s}=(1,1,3,2,5,3,\dots ,2n-1,n,n+1)$ (resp. $\mathbf{s}=(1,1,3,2,5,3,\dots ,2n-1,n)$). We also derive equidistributions that involve some natural tree statistics among weakly increasing trees on different multisets. Via introducing a group action on weakly increasing trees, we prove combinatorially the γ-positivity of the M-Eulerian–Narayana polynomials for a general multiset M. As an application of this γ-positivity result, we obtain a combinatorial interpretation of γ-coefficients of descent polynomials of permutations on the multiset $\{1^2,2^2,\dots ,n^2\}$ in terms of weakly increasing trees.

在本文中，我们介绍了在多集M上弱增加树的概念，它是平面树的扩展，并且在集合上增加树$\{0，\mathrm{1个}，\dots ，ñ\}$。我们为多集M上的弱增长树定义M -Eulerian-Narayana多项式，该多项式插值在Eulerian多项式和Narayana多项式之间。我们为通用多集上的弱增长树的数量获得了一个紧凑的产品公式。受多重置换和s-反转序列之间一些显着的均等分布的启发，我们为多重集建立了M -Eulerian-Narayana多项式之间的两个联系$\mathrm{中号}=\{{\mathrm{1个}}^{\mathrm{2个}}，{\mathrm{2个}}^{\mathrm{2个}}，\dots ，{ñ}^{\mathrm{2个}}\}$ （分别 $\mathrm{中号}=\{{\mathrm{1个}}^{\mathrm{2个}}，{\mathrm{2个}}^{\mathrm{2个}}，\dots ，{（ñ-\mathrm{1个}）}^{\mathrm{2个}}，ñ\}$）以及序列的Savage和Schuster的s -Eulerian多项式$\mathbf{s}=（\mathrm{1个}，\mathrm{1个}，3，\mathrm{2个}，5，3，\dots ，\mathrm{2个}ñ-\mathrm{1个}，ñ，ñ+\mathrm{1个}）$ （分别 $\mathbf{s}=（\mathrm{1个}，\mathrm{1个}，3，\mathrm{2个}，5，3，\dots ，\mathrm{2个}ñ-\mathrm{1个}，ñ）$）。我们还推导了在不同多集上的弱增长树之间涉及一些自然树统计信息的等分分布。通过在弱增长的树上引入群作用，我们组合证明了一般多集M的M -Eulerian-Narayana多项式的γ-正性。作为该γ阳性结果的应用，我们获得了多集置换的下降多项式的γ系数的组合解释。$\{{\mathrm{1个}}^{\mathrm{2个}}，{\mathrm{2个}}^{\mathrm{2个}}，\dots ，{ñ}^{\mathrm{2个}}\}$ 就树木的微弱增长而言。