Abstract
For a positive integer d, a non-negative integer n and a non-negative integer hle 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.
Highlights
Introduction and description of the resultsEnumeration is often the starting point in understanding a given mathematical structure
The motivation for the present paper comes from our attempt to understand the structure of d-tonal partition algebras using combinatorics of Green’s relations for the finite d-tonal partition monoid
This reduces the claim of the theorem to the following crucial observation: Ideals in tonal partition monoids
Summary
Enumeration is often the starting point in understanding a given mathematical structure. The motivation for the present paper comes from our attempt to understand the structure of d-tonal partition algebras using combinatorics of Green’s relations for the finite d-tonal partition monoid. As a corollary of this uniform description for all d, we obtain an alternative, simpler, description of A028289 using partitions with at most 3 parts
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