Crossings over permutations avoiding some pairs of patterns of length three. (English) Zbl 1442.05009
Summary: In this paper, we compute the distributions of the statistic number of crossings over permutations avoiding one of the pairs \(\{321, 231\}\), \(\{123, 132\}\) and \(\{123, 213\}\). The obtained results are new combinatorial interpretations of two known triangles in terms of restricted permutations statistic. For other pairs of patterns of length three, we find relationships between the polynomial distributions of the crossings over permutations that avoid the pairs containing the pattern 231 on the one hand, and the pattern 312 on the other hand.
MSC:
| 05A05 | Permutations, words, matrices |
| 05A19 | Combinatorial identities, bijective combinatorics |
| 05A15 | Exact enumeration problems, generating functions |
| 05A10 | Factorials, binomial coefficients, combinatorial functions |
Online Encyclopedia of Integer Sequences:
Triangle a(n,k) giving number of binary sequences of length n containing k subsequences 00.Number of permutations of length n that avoid the patterns 213 and 312 and have k double ascents, read by rows.
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