### Abstract

An involution 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 enumeration of involutions which avoid a set Ak of subsequences increasing both in number and in length at the same time. Let Ak be the set of all the permutations 12π3 . . . πk of length k. For k = 3 the only subsequence in Ak is 123 and the 123-avoiding involutions of length n are enumerated by the central binomial coefficients. For k = 4 we give a combinatorial explanation that shows the number of involutions of length n avoiding A4 is the same as the number of symmetric Schroder paths of length n − 1. For each k ≥ 3 we determine the generating function for the number of involutions avoiding the subsequences in Ak, according to length, first entry and number of fixed points.

Original language | English |
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Publication status | Published - 2007 |

Event | 19th International Conference on Formal Power Series & Algebraic Combinatorics - Nankai University, Tianjin, China Duration: 2 Jul 2007 → 6 Jul 2007 |

### Conference

Conference | 19th International Conference on Formal Power Series & Algebraic Combinatorics |
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Abbreviated title | FPSAC'07 |

Country | China |

City | Tianjin |

Period | 2/07/07 → 6/07/07 |

### Keywords

- involutions
- forbidden subsequences
- Schroder paths
- symmetric Schroder paths

## Cite this

Dukes, W. M. B., & Mansour, T. (2007).

*Involutions avoiding the class of permutations in Sk with prefix 12*. Paper presented at 19th International Conference on Formal Power Series & Algebraic Combinatorics, Tianjin, China.