Related papers: Restricted permutations, continued fractions, and …

Let R(n,k) be the number of permutations of $\{1,2,\ldots,n\}$ with k alternating runs. In this paper, we establish the relationships between R(n,k) and the central factorial numbers of even indices as well as the number of signed…

We count permutations avoiding a nonconsecutive instance of a two- or three-letter pattern, that is, the pattern may occur but only as consecutive entries in the permutation. Two-letter patterns give rise to the Fibonacci numbers. The…

In [BabStein] Babson and Steingr\'{\i}msson introduced generalized permutation patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. In [Kit1] Kitaev considered simultaneous…

A class P_{n,m,p}(x) of polynomials is defined. The combinatorial meaning of its coefficients is given. Chebyshev polynomials are the special cases of P_{n,m,p}(x). It is first shown that P_{n,m,p}(x) may be expressed in terms of…

We prove that every sufficiently long simple permutation contains two long almost disjoint simple subsequences. This result has applications to the enumeration of restricted permutations. For example, it immediately implies a result of Bona…

Gessel's famous Bessel determinant formula gives the generating function of the number of permutations without increasing subsequences of a given length. Ekhad and Zeilberger proposed the challenge of finding a suitable generalization for…

We construct an absolutely normal number whose continued fraction expansion is normal in the sense that it contains all finite patterns of partial quotients with the expected asymptotic frequency as given by the Gauss-Kuzmin measure. The…

The even and odd Zernike Polynomials R_n^m(x) can be expanded into sums of even and odd Chebyshev Polynomials T_i(x). This manuscript provides closed forms for the rational expansion coefficients c_{n,m,i} for a set of small 0 <= n-m <= 6…

In 1990 West conjectured that there are $2(3n)!/((n+1)!(2n+1)!)$ two-stack sortable permutations on $n$ letters. This conjecture was proved analytically by Zeilberger in 1992. Later, Dulucq, Gire, and Guibert gave a combinatorial proof of…

Define $S_n^k(\alpha)$ to be the set of permutations of $\{1,2,...,n\}$ with exactly $k$ fixed points which avoid the pattern $\alpha \in S_m$. Let $s_n^k(\alpha)$ be the size of $S_n^k(\alpha)$. We investigate $S_n^0(\alpha)$ for all…

Let F_{q^n} be the field of order q^n, and let Tr be the trace map from F_{q^n} to its q-element subfield. We exhibit nine sequences of polynomials of the form f(x):=x+c*Tr(x^k), with c in F_{q^n}, such that for each polynomial the function…

Recently, Bill Chen, together with his disciples Alvin Dai and Robin Zhou, discovered, and very elegantly proved, an algebraic equation satisfied by the generating function enumerating 123-avoiding words with two occurrences of each of 1,…

Recently, Babson and Steingrimsson (see \cite{BS}) introduced generalized permutations patterns that allow the requirement that two adjacent letters in a pattern must be adjacent in the permutation. In this paper we study the generating…

Using notions of composita and composition of generating functions we obtain explicit formulas for Chebyshev polynomials, Legendre polynomials, Gegenbauer polynomials, Associated Laguerre polynomials, Stirling polynomials, Abel polynomials,…

We study the longest increasing subsequence problem for random permutations avoiding the pattern $312$ and another pattern $\tau$ under the uniform probability distribution. We determine the exact and asymptotic formulas for the average…

In this paper, we consider cyclic permutations that avoid the monotone decreasing permutation $k(k-1)\ldots 21$, whose cycle also demonstrates some pattern avoidance. If the cycle is written in standard form with 1 appearing at the…

We describe a simple method that produces automatically closed forms for the coefficients of continued fractions expansions of a large number of special functions. The function is specified by a non-linear differential equation and initial…

For each prime p other than 3, and each power q=p^k, we present two large classes of permutation polynomials over F_{q^2} of the form X^r B(X^{q-1}) which have at most five terms, where B(X) is a polynomial with coefficients in {1,-1}. The…

In this paper, we find an explicit formulas, or recurrences, in terms of generating functions for the cardinalities of the sets $S_n(T;\tau)$ of all permutations in $S_n$ that contain $\tau\in S_k$ exactly once and avoid a subset…

We give an improved algorithm for counting the number of $1324$-avoiding permutations, resulting in $14$ further terms of the generating function, which is now known for all patterns of length $\le 50$. We re-analyse the generating function…