On the number of principal ideals in d-tonal partition monoids

TL;DR

This paper investigates the enumeration of principal ideals in d-tonal partition monoids, deriving formulas and discovering new integral sequences related to lattice structures and partition combinatorics.

Contribution

It provides closed-form formulas for counting principal ideals and introduces new integral sequences linked to combinatorial and lattice-theoretic structures.

Findings

Closed-form formulas for principal ideals in d-tonal partition monoids

Introduction of new integral sequences from ideal counts

Connections established between sequences, lattices, and partition combinatorics

Abstract

For a positive integer $d$, a non-negative integer $n$ and a non-negative integer $h\leq n$, we study the number $C_{n}^{(d)}$ of principal ideals; and the number $C_{n,h}^{(d)}$ of principal ideals generated by an element of rank $h$, in the $d$-tonal partition monoid on $n$ elements. We compute closed forms for the first family, as partial cumulative sums of known sequences. The second gives an infinite family of new integral sequences. We discuss their connections to certain integral lattices as well as to combinatorics of partitions.