The M\"{o}bius Function of a Restricted Composition Poset

TL;DR

This paper investigates the Möbius function of a restricted composition poset, establishing its isomorphism to a word poset and deriving explicit formulas and rationality results using combinatorial and automata techniques.

Contribution

It introduces a new isomorphism between a composition poset and a word poset, enabling explicit Möbius function formulas and proving rationality of associated generating functions.

Findings

Möbius function formula: (u,w)=(-1)^{|u|+|w|}{wu}_{dn}

Both zeta and Möbius functions have rational generating series

The composition poset is isomorphic to the word poset A_d^*

Abstract

We study a poset of compositions restricted by part size under a partial ordering introduced by Bj\"{o}rner and Stanley. We show that our composition poset $C_{d+1}$ is isomorphic to the poset of words $A_d^*$. This allows us to use techniques developed by Bj\"{o}rner to study the M\"{o}bius function of $C_{d+1}$. We use counting arguments and shellability as avenues for proving that the M\"{o}bius function is $\mu(u,w)=(-1)^{|u|+|w|}{w\choose u}_{dn}$, where ${w\choose u}_{dn}$ is the number of $d$-normal embeddings of $u$ in $w$. We then prove that the formal power series whose coefficients are given by the zeta and the M\"{o}bius functions are both rational. Following in the footsteps of Bj\"{o}rner and Reutenauer and Bj\"{o}rner and Sagan, we rely on definitions to prove rationality in one case, and in another case we use finite-state automata.