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%I #81 Mar 02 2024 13:54:11
%S 1,1,3,17,151,1899,31711,672697,17551323,549500451,20246665349,
%T 864261579999,42190730051687,2329965898878307,144220683681814515,
%U 9926440976428215117,754465679498026783923,62939664181821196179459,5732069150321309755351161,567176164248814234096702451
%N Number of pairs of permutations of degree n that avoid (12,21).
%C A pair of permutations (p,q) of degree n avoid (12,21) if there do not exist indices 1<=i<j<=n such that p_i < p_j and q_j < q_i. - _Noam Zeilberger_, Jun 06 2016 (via Steve Linton)
%C Number of intervals (i.e. ordered pairs (x,y) such that x<=y) in the permutation lattice of size n, that is, pairs of permutations (x,y) related by the weak Bruhat order x<=y iff inversions(x) is a subset of inversions(y) (see Hammett and Pittel, p. 4567). - _Noam Zeilberger_, Jun 01 2016
%H Andrew Elvey Price, <a href="/A007767/b007767.txt">Table of n, a(n) for n = 0..26</a>
%H Noga Alon, Kirill Rudov, and Leeat Yariv, <a href="https://lyariv.mycpanel.princeton.edu/papers/DominanceSolvability.pdf">Dominance Solvability in Random Games</a>, Princeton Univ. (2020).
%H Noga Alon, Kirill Rudov, and Leeat Yariv, <a href="https://lyariv.mycpanel.princeton.edu/papers/DominanceSolvabilityAppendix.pdf">Online Appendix for 'Dominance Solvability in Random Games'</a>, Princeton Univ. (2021).
%H Grégory Chatel, Vincent Pilaud, and Viviane Pons, <a href="https://arxiv.org/abs/1701.07995">The weak order on integer posets</a>, arXiv:1701.07995 [math.CO], 2017.
%H Clément Chenevière, <a href="https://theses.hal.science/tel-04255439">Enumerative study of intervals in lattices of Tamari type</a>, Ph. D. thesis, Univ. Strasbourg (France), Ruhr-Univ. Bochum (Germany), HAL tel-04255439 [math.CO], 2024. See pp. 3, 33, 145.
%H Joël Gay and Vincent Pilaud, <a href="https://arxiv.org/abs/1804.06572">The weak order on Weyl posets</a>, arXiv:1804.06572 [math.CO], 2018.
%H Benjamin Gunby, <a href="https://arxiv.org/abs/1609.06023">Asymptotics of Pattern Avoidance in the Permutation-Tuple and Klazar Set Partition Settings</a>, arXiv:1609.06023 [math.CO], 2017.
%H Adam Hammett and Boris Pittel, <a href="http://dx.doi.org/10.1090/S0002-9947-08-04478-4">How often are two permutations comparable?</a>, Transactions of the AMS 360:9 (2008), 4541-4568.
%H Evgeny Kapun, <a href="/A007767/a007767.java.txt">Java program for generating terms a(0)-a(13)</a>.
%F a(n) = Sum_{k=1..n!} k * A263754(n,k). - _Alois P. Heinz_, Jun 06 2016
%o (Java) See link.
%Y Cf. A000260, A263754.
%K nonn
%O 0,3
%A _Steve Linton_
%E a(0)=1 prepended by _Alois P. Heinz_, Jun 06 2016
%E a(10)-a(13) from _Evgeny Kapun_, Dec 11 2016
%E More terms from _Andrew Elvey Price_, Feb 08 2024
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