On multi-avoidance of generalized patterns

In [Kit1] Kitaev discussed simultaneous avoidance of two 3-patterns with no internal dashes, that is, where the patterns correspond to contiguous subwords in a permutation. In three essentially different cases, the numbers of such $n$-permutations are $2^{n-1}$, the number of involutions in $\mathcal{S}_n$, and $2E_n$, where $E_n$ is the $n$-th Euler number. In this paper we give recurrence relations for the remaining three essentially different cases. To complete the descriptions in [Kit3] and [KitMans], we consider avoidance of a pattern of the form $x-y-z$ (a classical 3-pattern) and beginning or ending with an increasing or decreasing pattern. Moreover, we generalize this problem: we demand that a permutation must avoid a 3-pattern, begin with a certain pattern and end with a certain pattern simultaneously. We find the number of such permutations in case of avoiding an arbitrary generalized 3-pattern and beginning and ending with increasing or decreasing patterns.Comment: 26 page

The problem of the pawns

In this paper we study the number $M_{m,n}$ of ways to place nonattacking pawns on an $m\times n$ chessboard. We find an upper bound for $M_{m,n}$ and analyse its asymptotic behavior. It turns out that $\lim_{m,n\to\infty}(M_{m,n})^{1/mn}$ exists and is bounded from above by $(1+\sqrt{5})/2$. Also, we consider a lower bound for $M_{m,n}$ by reducing this problem to that of tiling an $(m+1)\times (n+1)$ board with square tiles of size $1\times 1$ and $2\times 2$. Moreover, we use the transfer-matrix method to implement an algorithm that allows us to get an explicit formula for $M_{m,n}$ for given $m$.Comment: 16 pages; 6 figure

Simultaneous avoidance of generalized patterns

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 avoidance (multi-avoidance) of two or more 3-patterns with no internal dashes, that is, where the patterns correspond to contiguous subwords in a permutation. There either an explicit or a recursive formula was given for all but one case of simultaneous avoidance of more than two patterns. In this paper we find the exponential generating function for the remaining case. Also we consider permutations that avoid a pattern of the form $x-yz$ or $xy-z$ and begin with one of the patterns $12... k$, $k(k-1)... 1$, $23... k1$, $(k-1)(k-2)... 1k$ or end with one of the patterns $12... k$, $k(k-1)... 1$, $1k(k-1)... 2$, $k12... (k-1)$. For each of these cases we find either the ordinary or exponential generating functions or a precise formula for the number of such permutations. Besides we generalize some of the obtained results as well as some of the results given in [Kit3]: we consider permutations avoiding certain generalized 3-patterns and beginning (ending) with an arbitrary pattern having either the greatest or the least letter as its rightmost (leftmost) letter.Comment: 18 page