|
%I #33 Dec 08 2021 07:35:32
%S 1,1,1,1,4,6,456,6240,10936320,1225566720,130025295912960,
%T 252282619805368320,2209617218725251404267520,
%U 98758655816833727741338583040
%N Number of n X n symmetric matrices whose first row is 1..n and whose rows and columns are all permutations of 1..n.
%C The odd subsequence is A000438. The even subsequence is A035483.
%H Brendan D. McKay and Ian M. Wanless, <a href="https://doi.org/10.1002/jcd.21814">Enumeration of Latin squares with conjugate symmetry</a>, J. Combin. Des. 30 (2022), 105-130, also on <a href="https://arxiv.org/abs/2104.07902">Enumeration of Latin squares with conjugate symmetry</a>, arXiv:2104.07902 [math.CO], 2021. Table 2 p. 7.
%e 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.
%t (* 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 *)
%Y Cf. A000438, A035482, A000315, A002860, A003090, A040082.
%K nonn,more,nice
%O 0,5
%A _Joshua Zucker_ and Joe Keane
%E a(10)-a(13) from _Ian Wanless_, Oct 20 2019
|
