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

1, 1, 4, 35, 545, 13250, 463899, 22106253, 1375915620, 108386009099, 10540705282001, 1240370638524842, 173704235075714947, 28549174106487593365, 5441843626292088857818, 1190762128123996264128849, 296456799935194225886732961, 83321234634397591315509479058

Offset: 0

Author

Alois P. Heinz, Apr 27 2020

Keywords

• nonn

Examples

			a(3) = (3!)^2 - 1 = 35: only (321,321) does not avoid synchronous double descent pairs among the ordered pairs of permutations of [3].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..100
• Wikipedia, Permutation

Crossrefs

Column k=0 of A334257.

Cf. A000275 (the same for single descent pairs), A001044.

Programs

Maple

b:= proc(n, u, v, t) option remember; `if`(n=0, 1, add(add(
      b(n-1, sort([u-j, v-i])[], 1), i=1..v)+add(
      b(n-1, sort([u-j, v+i-1])[], 1), i=1..n-v), j=1..u)+add(add(
      b(n-1, sort([u+j-1, v-i])[], 1), i=1..v)+add(`if`(t=0, 0,
      b(n-1, sort([u+j-1, v+i-1])[], 0)), i=1..n-v), j=1..n-u))
    end:
a:= n-> b(n$3, 1):
seq(a(n), n=0..21);

Mathematica

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 &] &,
Range[0, nn]!^2 CoefficientList[Series[1/(1 - x - a), {x, 0, nn}], {x, y}]] // Flatten (* Geoffrey Critzer, Apr 27 2020 *)

Formula

a(n) <= A001044(n) with equality only for n < 3.