Classifying Descents According to Parity

Classifying Descents According to equivalence mod k

In [S. Kitaev and J. Remmel: Classifying descents according to parity] the authors refine the well-known permutation statistic "descent" by fixing parity of (exactly) one of the descent's numbers. In this paper, we generalize the results of [S. Kitaev and J. Remmel: Classifying descents according to parity] by studying descents according to whether the first or the second element in a descent pair is equivalent to $k$ mod $k\geq 2$. We provide either an explicit or an inclusion-exclusion type formula for the distribution of the new statistics. Based on our results we obtain combinatorial proof…

按下降位置奇偶性的排列计数

Enumeration of permutations by the parity of descent positions

本研究发现近期部分下降多项式变体属Carlitz-Scoville四变量多项式特例，证明了该生成函数的q-模拟与B型模拟，还得到Chebikin公式的q-模拟、Sun二元欧拉多项式伽马正性新证及细化Petersen结果的B型模拟。

基于降数奇偶性的$q$-枚举B和D型欧拉多项式

$q$-enumeration of type B and D Eulerian polynomials based on parity of descents

本文研究了基于降数奇偶性的B和D型欧拉多项式$q$-枚举问题，通过引入四种统计量和B型翻转数，推导了相关生成函数及递推关系，同时给出了D型Coxeter群的五变量多项式及其指数生成函数。

Counting descents, rises, and levels, with prescribed first element, in words

Recently, Kitaev and Remmel [Classifying descents according to parity, Annals of Combinatorics, to appear 2007] refined the well-known permutation statistic ``descent'' by fixing parity of one of the descent's numbers. Results in that paper were extended and generalized in several ways. In this paper, we shall fix a set partition of the natural numbers $N$, $(N_1, ..., N_t)$, and we study the distribution of descents, levels, and rises according to whether the first letter of the descent, rise, or level lies in $N_i$ over the set of words over the alphabet $[k]$. In particular, we refine and g…

Descents in $t$-Sorted Permutations

Let $s$ denote West's stack-sorting map. A permutation is called $t-\textit{sorted}$ if it is of the form $s^t(μ)$ for some permutation $μ$. We prove that the maximum number of descents that a $t$-sorted permutation of length $n$ can have is $\left\lfloor\frac{n-t}{2}\right\rfloor$. When $n$ and $t$ have the same parity and $t\geq 2$, we give a simple characterization of those $t$-sorted permutations in $S_n$ that attain this maximum. In particular, the number of such permutations is $(n-t-1)!!$.

受限排列上的奇偶统计量与Catalan–Schett多项式

Parity statistics on restricted permutations and the Catalan--Schett polynomials

受可栈排序排列与Jacobi椭圆函数研究启发，本文构造新双射并细化$321$-避免排列的下降生成函数，解决了Kitaev和Zhang关于非重叠上升的公开问题，给出Catalan–Schett多项式的组合解释。