05960069

j

05960069 Burstein, Alexander A short proof for the number of permutations containing pattern 321 exactly once. Electron. J. Comb. 18, No. 2, Research Paper P21, 3 p., electronic only (2011). 2011 Prof. Andr\'e K\"undgen, Deptartment of Mathematics, California State University San Marcos, San Marcos, CA EN Zbl 0852.05009 Zbl 1022.05005

emis:journals/EJC/Volume_18/Abstracts/v18i2p21.html

A {\it pattern\/} is an equivalence class of sequences under order isomorphism; a sequence $\sigma$ {\it contains\/} a pattern $\tau$ if $\sigma$ has a subsequence which is order-isomorphic to $\tau$. The set of permutations in $S_n$ containing pattern $\tau$ exactly $r$ times is denoted by $S_n(\tau;r)$. The author provides a ``short" proof of {\bf Theorem 3}: $\left|S_n(321;1)\right|=\frac3n\cdot{\binom{2n}{n-3}}$ [{\it J. Noonan}, ``The number of permutations containing exactly one increasing subsequence of length three", Discrete Math.\ 152, No.\ 1--3, 307--313 (1996; Zbl 0852.05009)], using the ``block decomposition method" of [{\it T. Mansour} and {\it A. Vainshtein}, ``Restricted permutations and Chebyshev polynomials", S\'emin.\ Lothar.\ Comb.\ 47, 17 p.\ (2002; Zbl 1022.05005)]. William G. Brown (Montr\'eal)