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%I #36 Nov 27 2023 16:20:28
%S 1,1,0,1,1,0,0,4,0,0,0,5,3,0,0,0,2,14,0,0,0,0,0,23,9,0,0,0,0,0,16,48,
%T 0,0,0,0,0,0,4,97,27,0,0,0,0,0,0,0,94,162,0,0,0,0,0,0,0,0,44,387,81,0,
%U 0,0,0,0,0,0,0,8,476,540,0,0,0,0,0,0,0,0,0,0,320,1485,243,0,0,0,0,0,0
%N Triangle read by rows: T(n,k) is the number of permutations of length n avoiding simultaneously the patterns 123 and 132 with the maximum number of non-overlapping descents equal k.
%C Number of permutations of length n avoiding simultaneously the patterns 123 and 132 with the maximum number of non-overlapping descents equal k. A descent in a permutation a(1)a(2)...a(n) is position i such that a(i) > a(i+1).
%H Tian Han and Sergey Kitaev, <a href="https://arxiv.org/abs/2311.02974">Joint distributions of statistics over permutations avoiding two patterns of length 3</a>, arXiv:2311.02974 [math.CO], 2023. See formula 7 at page 7.
%F G.f.: (1 + x + x^2 - 2*x^2*z - x^3*z)/(1 - 3*x^2*z - 2*x^3*z).
%e Triangle T(n,k) begins:
%e 1;
%e 1, 0;
%e 1, 1, 0;
%e 0, 4, 0, 0;
%e 0, 5, 3, 0, 0;
%e 0, 2, 14, 0, 0, 0;
%e 0, 0, 23, 9, 0, 0, 0;
%e 0, 0, 16, 48, 0, 0, 0, 0;
%e 0, 0, 4, 97, 27, 0, 0, 0, 0;
%e 0, 0, 0, 94, 162, 0, 0, 0, 0, 0;
%e 0, 0, 0, 44, 387, 81, 0, 0, 0, 0, 0;
%e 0, 0, 0, 8, 476, 540, 0, 0, 0, 0, 0, 0;
%e 0, 0, 0, 0, 320, 1485, 243, 0, 0, 0, 0, 0, 0;
%e ...
%Y Row sums give A011782.
%Y Column sums give 3*A005054.
%Y T(2n,n) gives A133494.
%Y T(3n+2,n) gives A000079.
%Y T(3n+1,n) gives A053220(n+1).
%K nonn,tabl
%O 0,8
%A _Tian Han_, Nov 24 2023
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