%I #5 Nov 15 2018 21:12:34

%S 1,1,1,13,13,45,681,885,2805,8301,237213

%N Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, with distinct row sums and distinct column sums.

%e The a(3) = 13 matrices:

%e [3] [2 1] [1 2]

%e .

%e [2] [2 0] [1 1] [1 1] [1] [1 0] [1 0] [0 2] [0 1] [0 1]

%e [1] [0 1] [1 0] [0 1] [2] [1 1] [0 2] [1 0] [2 0] [1 1]

%t prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];

%t multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];

%t Table[Length[Select[multsubs[Tuples[Range[n],2],n],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],UnsameQ@@Total/@prs2mat[#],UnsameQ@@Total/@Transpose[prs2mat[#]]]&]],{n,5}]

%Y Cf. A000219, A001970, A007716, A068313, A114736, A117433, A120733, A319646, A321645, A321646, A321652.

%K nonn,more

%O 0,4

%A _Gus Wiseman_, Nov 15 2018