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%I #18 Jan 24 2022 17:05:41
%S 1,2,10,60,388,2606,17890,124512,874562,6182198,43903044,312843918,
%T 2235028210,15999423988,114710881886,823463493632,5917220509358
%N Number of permutations avoiding 321 of length 3n composed of only 3-cycles.
%C Sum over all Dyck paths D of L(D)*2^h(D), where h(D) is the number of times the Dyck path hits the x-axis and L(D) is the product of binomial coefficients (u_i(D)+d_i(D) choose u_i(D)), where u_i(D) is the number of up-steps between the i-th and (i+1)-st down step and d_i(D) is the number of down-steps between the i-th and (i+1)-st up step.
%H Kassie Archer and Christina Graves, <a href="https://arxiv.org/abs/2104.12664">Pattern-restricted permutations composed of 3-cycles</a>, arXiv:2104.12664 [math.CO], 2021.
%e For n=2, the ten permutations (in one-line notation and cycle notation) are:
%e [2, 3, 1, 5, 6, 4] (1,2,3)(4,5,6)
%e [3, 1, 2, 5, 6, 4] (1,3,2)(4,5,6)
%e [2, 3, 1, 6, 4, 5] (1,2,3)(4,6,5)
%e [3, 1, 2, 6, 4, 5] (1,3,2)(4,6,5)
%e [4, 1, 6, 2, 3, 5] (1,4,2)(3,6,5)
%e [2, 4, 6, 1, 3, 5] (1,2,4)(3,6,5)
%e [4, 1, 5, 2, 6, 3] (1,4,2)(3,5,6)
%e [5, 6, 1, 2, 3, 4] (1,5,3)(2,6,4)
%e [2, 4, 5, 1, 6, 3] (1,2,4)(3,5,6)
%e [3, 4, 5, 6, 1, 2] (1,3,5)(2,4,6)
%Y Cf. A350645.
%K nonn,more
%O 0,2
%A _Kassie Archer_, Jan 10 2022
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