225 research outputs found
The M\"obius function of the consecutive pattern poset
An occurrence of a consecutive permutation pattern in a permutation
is a segment of consecutive letters of whose values appear in the same
order of size as the letters in . The set of all permutations forms a poset
with respect to such pattern containment. We compute the M\"obius function of
intervals in this poset, providing what may be called a complete solution to
the problem. For most intervals our results give an immediate answer to the
question. In the remaining cases, we give a polynomial time algorithm to
compute the M\"obius function. In particular, we show that the M\"obius
function only takes the values -1, 0 and 1.Comment: 10 pages, 2 figure
Restricted binary strings and generalized Fibonacci numbers
Part 2: Regular PapersInternational audienceWe provide some interesting relations involving k-generalized Fibonacci numbers between the set of length n binary strings avoiding k of consecutive 0’s and the set of length n strings avoiding consecutive 0’s and 1’s with some more restriction on the first and last letter, via a simple bijection. In the special case a probably new interpretation of Fibonacci numbers is given.Moreover, we describe in a combinatorial way the relation between the strings of with an odd numbers of 1’s and the ones with an even number of 1’s
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