Euler characteristic of the truncated order complex of generalized noncrossing partitions

A combinatorial bijection on $k$-noncrossing partitions

For any integer $k\geq2$, we prove combinatorially the following Euler (binomial) transformation identity $$ \NC_{n+1}^{(k)}(t)=t\sum_{i=0}^n{n\choose i}\NW_{i}^{(k)}(t), $$ where $\NC_{m}^{(k)}(t)$ (resp.~$\NW_{m}^{(k)}(t)$) is the sum of weights, $t^\text{number of blocks}$, of partitions of $\{1,\ldots,m\}$ without $k$-crossings (resp.~enhanced $k$-crossings). The special $k=2$ and $t=1$ case, asserting the Euler transformation of Motzkin numbers are Catalan numbers, was discovered by Donaghey 1977. The result for $k=3$ and $t=1$, arising naturally in a recent study of pattern avoidance in …

非交叉划分与碎片交集序

Noncrossing partitions and the shard intersection order

本文定义了有限Coxeter群W元素的新格结构，即碎片交集序，其弱序更弱，并包含非交叉划分格NC(W)。该构造提供了NC(W)为格的新证明，且其格属性与NC(W)相似。

Truncated Series with Nonnegative Coefficients from the Jacobi Triple Product

Andrews and Merca investigated a truncated version of Euler's pentagonal number theorem and showed that the coefficients of the truncated series are nonnegative. They also considered the truncated series arising from Jacobi's triple product identity, and they that its coefficients are nonnegative. This conjecture was posed by Guo and Zeng independently and confirmed by Mao and Yee using different approaches. In this paper, we provide a new combinatorial proof of their nonnegativity result related to Euler's pentagonal number theorem. Meanwhile, we find an analogous result for a truncated serie…

来自Bailey格的截断theta级数

Truncated theta series from the Bailey lattice

本研究利用Bailey格推导奇基雅可比三重积级数的截断形式，得到多个经典theta级数的新截断式与分拆函数不等式，同时证实了相关分拆猜想并证明Merca强猜想在R=3S（S≥1）时成立。

Truncated theta series and partitions into distinct parts

Linear inequalities involving Euler's partition function $p(n)$ have been the subject of recent studies. In this article, we consider the partition function $Q(n)$ counting the partitions of $n$ into distinct parts. Using truncated theta series, we provide four infinite families of linear inequalities for $Q(n)$ and partition theoretic interpretations for these results.

Lecture Hall Theorems, q-series and Truncated Objects

We show here that the refined theorems for both lecture hall partitions and anti-lecture hall compositions can be obtained as straightforward consequences of two q-Chu Vandermonde identities, once an appropriate recurrence is derived. We use this approach to get new lecture hall-type theorems for truncated objects. We compute their generating function and give two different multivariate refinements of these new results : the q-calculus approach gives (u,v,q)-refinements, while a completely different approach gives odd/even (x,y)-refinements. From this, we are able to give a combinatorial chara…