Layered restrictions and Chebyshev polynomials

Involutions Restricted by 3412, Continued Fractions, and Chebyshev Polynomials

We study generating functions for the number of involutions, even involutions, and odd involutions in $S_n$ subject to two restrictions. One restriction is that the involution avoid 3412 or contain 3412 exactly once. The other restriction is that the involution avoid another pattern $τ$ or contain $τ$ exactly once. In many cases we express these generating functions in terms of Chebyshev polynomials of the second kind.

Restricted permutations and Chebyshev polynomials

We study generating functions for the number of permutations in $\SS_n$ subject to two restrictions. One of the restrictions belongs to $\SS_3$, while the other to $\SS_k$. It turns out that in a large variety of cases the answer can be expressed via Chebyshev polynomials of the second kind.

Restricted set of patterns, continued fractions, and Chebyshev polynomials

We study generating functions for the number of permutations in $S_n$ subject to set of restrictions. One of the restrictions belongs to $S_3$, while the others to $S_k$. It turns out that in a large variety of cases the answer can be expressed via continued fractions, and Chebyshev polynomials of the second kind.

Restricted Motzkin permutations, Motzkin paths, continued fractions, and Chebyshev polynomials

We say that a permutation $π$ is a Motzkin permutation if it avoids 132 and there do not exist $a<b$ such that $π_a<π_b<π_{b+1}$. We study the distribution of several statistics in Motzkin permutations, including the length of the longest increasing and decreasing subsequences and the number of rises and descents. We also enumerate Motzkin permutations with additional restrictions, and study the distribution of occurrences of fairly general patterns in this class of permutations.

Forbidding intersection patterns between layers of the cube

A family ${\mathcal A} \subset {\mathcal P} [n]$ is said to be an antichain if $A \not \subset B$ for all distinct $A,B \in {\mathcal A}$. A classic result of Sperner shows that such families satisfy $|{\mathcal A}| \leq \binom {n}{\lfloor n/2\rfloor}$, which is easily seen to be best possible. One can view the antichain condition as a restriction on the intersection sizes between sets in different layers of ${\mathcal P} [n]$. More generally one can ask, given a collection of intersection restrictions between the layers, how large can families respecting these restrictions be? Answering a que…

分层排列的堆积密度及分层排列中单调序列的最小数量

Packing densities of layered permutations and the minimum number of monotone sequences in layered permutations

本文提出两个新结果：推广定理计算特定分层排列的堆积密度，并证明任意大分层排列中长度为k+1的单调序列最小密度渐近为1/k^k，与Myers猜想相容。