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Triangle read by rows: T(n,k) is the number of permutations p of [n] in which the length of the longest initial segment avoiding both the 132- and the 321-pattern is equal to k.
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%I #6 Sep 18 2017 13:06:41

%S 1,0,2,0,2,4,0,8,9,7,0,40,45,24,11,0,240,270,144,50,16,0,1680,1890,

%T 1008,350,90,22,0,13440,15120,8064,2800,720,147,29,0,120960,136080,

%U 72576,25200,6480,1323,224,37,0,1209600,1360800,725760,252000,64800,13230

%N Triangle read by rows: T(n,k) is the number of permutations p of [n] in which the length of the longest initial segment avoiding both the 132- and the 321-pattern is equal to k.

%C Row sums are the factorial numbers (A000142).

%C T(n,2)=n!/3 for n>=3 (A002301). T(n,3)=3n!/8 for n>=4.

%C Diagonal yields A000124.

%H E. Deutsch and W. P. Johnson, <a href="http://www.jstor.org/stable/3219101">Create your own permutation statistics</a>, Math. Mag., 77, 130-134, 2004.

%H R. Simion and F. W. Schmidt, <a href="https://doi.org/10.1016/S0195-6698(85)80052-4">Restricted permutations</a>, European J. Combin., 6, 383-406, 1985.

%F T(n, k) = n!k/[2(k-2)!(k+1)] for k<n; T(n, n) = n(n-1)/2.

%e T(3,2)=2 because only 132 and 321 satisfy the requirements.

%Y Cf. A000142, A002301, A000124.

%K nonn,tabl

%O 1,3

%A _Emeric Deutsch_, Apr 12 2004