On sums of figurate numbers by using techniques of poset representation theory

Figurate numbers and sums of powers of integers

Recently, Marko and Litvinov (ML) conjectured that, for all positive integers $n$ and $p$, the $p$-th power of $n$ admits the representation $n^p = \sum_{\ell =0}^{p-1} (-1)^{l} c_{p,\ell} F_{n}^{p-\ell}$, where $F_{n}^{p-\ell}$ is the $n$-th hyper-tetrahedron number of dimension $p-\ell$ and $c_{p,\ell}$ denotes the number of $(p -\ell)$-dimensional facets formed by cutting the $p$-dimensional cube $0 \leq x_1, x_2, \ldots, x_p \leq n-1$. In this paper we show that the ML conjecture is true for every natural number $p$. Our proof relies on the fact that the validity of the ML conjecture neces…

Partially Ordinal Sums and $P$-partitions

We present a method of computing the generating function $f_P(\x)$ of $P$-partitions of a poset $P$. The idea is to introduce two kinds of transformations on posets and compute $f_P(\x)$ by recursively applying these transformations. As an application, we consider the partially ordinal sum $P_n$ of $n$ copies of a given poset, which generalizes both the direct sum and the ordinal sum. We show that the sequence $\{f_{P_n}(\x)\}_{n\ge 1}$ satisfies a finite system of recurrence relations with respect to $n$. We illustrate the method by several examples, including a kind of 3-rowed posets and the…

整除关系下三角形数偏序集的默比乌斯函数

The Möbius function of the poset of triangular numbers under divisibility

本文研究了整除关系下三角形数偏序集上的默比乌斯函数，通过Hasse图和矩阵可视化分析其部分和与经典默比乌斯函数的异同，并提出了关于其渐近行为和上下界的新猜想。