The monoid representation of upho posets and total positivity

TL;DR

This paper characterizes totally positive formal power series with integer coefficients as rank-generating functions of specific posets, connecting poset theory with monoid structures and resolving a key conjecture.

Contribution

It establishes a bijection between upho posets and certain monoids, and introduces semi-upho posets and convolution operations within this framework.

Findings

All totally positive formal power series with integer coefficients and constant 1 are rank-generating functions of upho posets.

A bijection between finitary colored upho posets and atomic, left-cancellative monoids is constructed.

Introduction of semi-upho posets and a convolution operation on colored upho posets.

Abstract

We show that all totally positive formal power series with integer coefficients and constant term $1$ are precisely the rank-generating functions of Schur-positive upho posets, thereby resolving the main conjecture proposed by Gao, Guo, Seetharaman, and Seidel. To achieve this, we construct a bijection between finitary colored upho posets and atomic, left-cancellative, invertible-free monoids, which restricts to a correspondence between $\mathbb{N}$-graded colored upho posets and left-cancellative homogeneous monoids. Furthermore, we introduce semi-upho posets and develop a convolution operation on colored upho posets with colored semi-upho posets within this monoid-theoretic framework.