A short proof for the number of permutations containing pattern 321 exactly once (Q640473)

A \textit{pattern} is an equivalence class of sequences under order isomorphism; a sequence \(\sigma\) \textit{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 \textbf{Theorem 3}: \(\left|S_n(321;1)\right|=\frac3n\cdot{\binom{2n}{n-3}}\) [\textit{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 [\textit{T. Mansour} and \textit{A. Vainshtein}, ``Restricted permutations and Chebyshev polynomials'', Sémin.\ Lothar.\ Comb.\ 47, 17 p.\ (2002; Zbl 1022.05005)].