Abstract
We develop a method for counting words subject to various restrictions by finding a combinatorial interpretation for a product of weighted sums of Laguerre polynomials with parameter $\alpha = -1$. We describe how such a series can be computed by finding an appropriate ordinary generating function and applying a certain transformation. We use this technique to find the generating function for the number of $k$-ary words avoiding any vincular pattern that has only ones, as well as words cyclically avoiding vincular patterns with only ones whose runs of ones between dashes are all of equal length.
Highlights
We first consider the following simple problem
We develop a method for counting words subject to various restrictions by finding a combinatorial interpretation for a product of weighted sums of Laguerre polynomials with parameter α = −1
We describe how such a series can be computed by finding an appropriate ordinary generating function and applying a certain transformation. We use this technique to find the generating function for the number of k-ary words avoiding any vincular pattern that has only ones, as well as words cyclically avoiding vincular patterns with only ones whose runs of ones between dashes are all of equal length
Summary
We first consider the following simple problem. How many arrangements of the word “WALLAWALLA” are there with no LLL, AAA or WW as consecutive subwords? Perhaps surprisingly, the answer can be calculated by performing a certain integral. This formula involves the use of the maps T and Φ, but these can be calculated. We can use Sage to compute the the generating function W xlen(W ) where the sum is taken over all ternary words W avoiding the pattern 11-11, where len(W ) is the length of W , the number of letters counting multiplicity: x4 − 32 x3 + 24 x2 − 8 x + 1. This gives the generating functions for words so that any cyclic permutation of their letters avoids such a pattern. This generalizes a result of Burstein and Wilf [8] which gives the generating function for the number of words cyclically avoiding 1m
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