For a positive integer d, a non-negative integer n and a non-negative integer \(h\le n\), 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.

中文翻译：

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关于单调分区半体中的主要理想个数

对于一个正整数d，一个非负整数n和一个非负整数\（h \ le n \），我们研究本理理想数\（C_ {n} ^ {（d）} \）；和数量\（{C_ N，H} ^ {（d）} \）由秩的元件产生的主理想的ħ，在d上-tonal分区幺Ñ元件。我们计算第一个族的封闭形式，作为已知序列的部分累积和。第二个给出无限的新整数序列族。我们将讨论它们与某些整体网格以及分区组合的关系。