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A334412 Number of ordered pairs of permutations of [n] avoiding synchronous double descent pairs.

%I A334412 #17 Apr 27 2020 20:22:15
%S A334412 1,1,4,35,545,13250,463899,22106253,1375915620,108386009099,
%T A334412 10540705282001,1240370638524842,173704235075714947,
%U A334412 28549174106487593365,5441843626292088857818,1190762128123996264128849,296456799935194225886732961,83321234634397591315509479058
%N A334412 Number of ordered pairs of permutations of [n] avoiding synchronous double descent pairs.
%H A334412 Alois P. Heinz, <a href="/A334412/b334412.txt">Table of n, a(n) for n = 0..100</a>
%H A334412 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A334412 a(n) <= A001044(n) with equality only for n < 3.
%e A334412 a(3) = (3!)^2 - 1 = 35: only (321,321) does not avoid synchronous double descent pairs among the ordered pairs of permutations of [3].
%p A334412 b:= proc(n, u, v, t) option remember; `if`(n=0, 1, add(add(
%p A334412       b(n-1, sort([u-j, v-i])[], 1), i=1..v)+add(
%p A334412       b(n-1, sort([u-j, v+i-1])[], 1), i=1..n-v), j=1..u)+add(add(
%p A334412       b(n-1, sort([u+j-1, v-i])[], 1), i=1..v)+add(`if`(t=0, 0,
%p A334412       b(n-1, sort([u+j-1, v+i-1])[], 0)), i=1..n-v), j=1..n-u))
%p A334412     end:
%p A334412 a:= n-> b(n$3, 1):
%p A334412 seq(a(n), n=0..21);
%t A334412 nn = 20; a=Apply[Plus,Table[Normal[Series[y x^3/(1 - y x - y x^2), {x, 0, nn}]][[n]]/(n +2)!^2, {n, 1, nn - 2}]] /. y -> -1;Map[Select[#, # > 0 &] &,
%t A334412 Range[0, nn]!^2 CoefficientList[Series[1/(1 - x - a), {x, 0, nn}], {x, y}]] // Flatten (* _Geoffrey Critzer_, Apr 27 2020 *)
%Y A334412 Column k=0 of A334257.
%Y A334412 Cf. A000275 (the same for single descent pairs), A001044.
%K A334412 nonn
%O A334412 0,3
%A A334412 _Alois P. Heinz_, Apr 27 2020