Abstract
A permutation π is said to be τ -avoiding if it does not contain any subsequence having all the same pairwise comparisons as τ . This paper concerns the characterization and enumeration of permutations which avoid a set F^j of subsequences increasing both in number and in length at the same time. Let F^j be the set of subsequences of the form σ (j+1)(j+2), σ being any permutation on \1,...,j\. For j=1 the only subsequence in F^1 is 123 and the 123-avoiding permutations are enumerated by the Catalan numbers; for j=2 the subsequences in F^2 are 1234 2134 and the (1234,2134)avoiding permutations are enumerated by the Schröder numbers; for each other value of j greater than 2 the subsequences in F^j are j! and their length is (j+2) the permutations avoiding these j! subsequences are enumerated by a number sequence \a_n\ such that C_n ≤ a_n ≤ n!, C_n being the nth Catalan number. For each j we determine the generating function of permutations avoiding the subsequences in F^j according to the length, to the number of left minima and of non-inversions.
Highlights
The study of permutations represents an interesting and relevant discipline in Mathematics which began with Euler who first analyzed permutation statistics related to the study of parameters different from their length [16]
The new problems coming from Computer Science led to the development of the concept of permutations with forbidden subsequences
✏ ✝✓ we study the class of permutations ✔ ✔ ✖✕ ✘✗ ❄ ✒✑ ❃ ✗ such that ✬ ✾ ✷❈❅ and any ✭✼✮ ✬ ✾ has the form ✭ ✷
Summary
The study of permutations represents an interesting and relevant discipline in Mathematics which began with Euler who first analyzed permutation statistics related to the study of parameters different from their length [16]. The new problems coming from Computer Science led to the development of the concept of permutations with forbidden subsequences. We translate the construction, represented by the generating tree, into formulae, obtaining a set of functional equations Their solution gives the generating function of the permutations according to the length, number of left minima, non-inversions and active sites. We are able to determine the generating function according to the length of the permutations, number of left minima and non-inversions. This result allows us to show that the generating function is algebraic of degree two, except for ❅✘✷
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