A370723 Number of symmetric (0,1)-matrices with sum of entries equal to n and no zero rows or columns, with weakly decreasing row sums and column sums.

1, 2, 5, 14, 39, 123, 393, 1352, 4782, 17824, 68481, 274166

Offset: 1

Author

Ludovic Schwob, May 18 2024

Keywords

• nonn
• more

Examples

			The a(3) = 5 matrices:
  [1 0 0]  [1 0 0]  [0 1 0]  [0 0 1]  [1 1]
  [0 1 0]  [0 0 1]  [1 0 0]  [0 1 0]  [1 0]
  [0 0 1]  [0 1 0]  [0 0 1]  [1 0 0]
		

Crossrefs

Cf. A178718, A135588, A321652, A365961.

Programs

Mathematica

a[0] = 1;
a[n_] := a[n] = Length[Select[Subsets[Tuples[Range[n], 2], {n}], Module[{matrix, rows, cols}, matrix = ConstantArray[0, {n, n}]; (matrix[[#[[1]], #[[2]]]] = 1) & /@ #; rows = Total[matrix, {2}]; cols = Total[matrix, {1}]; And[Union[First /@ #] == Range[Max @@ First /@ #], Union[Last /@ #] == Range[Max @@ Last /@ #], Sort[Reverse /@ #] == #, OrderedQ[Reverse[rows]], OrderedQ[Reverse[cols]]]] &]];
Table[a[n], {n, 1, 6}] (* Robert P. P. McKone, May 19 2024, from Gus Wiseman in A135588 *)