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

For a positive integer d, a non-negative integer n and a non-negative integer h <= 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.