A035481 Number of n X n symmetric matrices whose first row is 1..n and whose rows and columns are all permutations of 1..n.

1, 1, 1, 1, 4, 6, 456, 6240, 10936320, 1225566720, 130025295912960, 252282619805368320, 2209617218725251404267520, 98758655816833727741338583040

Offset

0,5

Comments

The odd subsequence is A000438. The even subsequence is A035483.

Links

Table of n, a(n) for n=0..13.

Brendan D. McKay and Ian M. Wanless, Enumeration of Latin squares with conjugate symmetry, J. Combin. Des. 30 (2022), 105-130, also on Enumeration of Latin squares with conjugate symmetry, arXiv:2104.07902 [math.CO], 2021. Table 2 p. 7.

Example

a(3) = 1 because after 123 in the first row and column, 213 is not allowed for the second row, so it must be 231 and thus the third row is 312.

Mathematica

(* This script is not suitable for n > 6 *) matrices[n_ /; n > 1] := Module[{a, t, vars}, t = Table[Which[i==1, j, j==1, i, j>i, a[i, j], True, a[j, i]], {i, n}, {j, n}]; vars = Select[Flatten[t], !IntegerQ[#]& ] // Union; t /. {Reduce[And @@ (1 <= # <= n & /@ vars) && And @@ Unequal @@@ t, vars, Integers] // ToRules}]; a[0] = a[1] = 1; a[n_] := Length[ matrices[n]]; Table[an = a[n]; Print["a(", n, ") = ", an]; an, {n, 0, 6}] (* Jean-François Alcover, Jan 04 2016 *)

Crossrefs

Cf. A000438, A035482, A000315, A002860, A003090, A040082.

Sequence in context: A113838 A056831 A027717 * A323214 A061214 A137024

Adjacent sequences: A035478 A035479 A035480 * A035482 A035483 A035484

Keyword

nonn,more,nice

Author

Joshua Zucker and Joe Keane

Extensions

a(10)-a(13) from Ian Wanless, Oct 20 2019

Status

approved