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Irregular triangle read by rows: T(n,k) = number of permutations of length n avoiding simultaneously the patterns 132 and 321 with the maximum number of non-overlapping ascents equal to k.
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%I #16 Dec 21 2023 21:12:10

%S 1,1,1,0,4,0,4,3,0,0,11,0,0,9,7,0,0,0,22,0,0,0,16,13,0,0,0,0,37,0,0,0,

%T 0,25,21,0,0,0,0,0,56,0,0,0,0,0,36,31,0,0,0,0,0,0,79,0,0,0,0,0,0,49,

%U 43,0,0,0,0,0,0,0,106

%N Irregular triangle read by rows: T(n,k) = number of permutations of length n avoiding simultaneously the patterns 132 and 321 with the maximum number of non-overlapping ascents equal to k.

%C An ascent 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.

%F G.f.: (1 + x + x^2 - 2*x^2*y + x^3*y + x^4*y + 3*x^4*y^2 + 2*x^5*y^2)/(1 - x^2*y)^3.

%e 1,

%e 1, 1,

%e 0, 4,

%e 0, 4, 3,

%e 0, 0, 11,

%e 0, 0, 9, 7,

%e 0, 0, 0, 22,

%e 0, 0, 0, 16, 13,

%e 0, 0, 0, 0, 37,

%e 0, 0, 0, 0, 25, 21,

%e 0, 0, 0, 0, 0, 56,

%e 0, 0, 0, 0, 0, 36, 31,

%e 0, 0, 0, 0, 0, 0, 79,

%e 0, 0, 0, 0, 0, 0, 49, 43,

%e 0, 0, 0, 0, 0, 0, 0, 106

%Y Cf. A367631.

%K nonn,tabf

%O 0,5

%A _Tian Han_, Dec 03 2023