A321660 Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, whose entries are all distinct.

1, 1, 1, 5, 5, 9, 45, 49, 85, 125, 233, 273, 417, 529, 745, 2573, 2861, 4761, 6837, 10489, 14317, 22637, 28289, 40041, 52041, 70177, 88561, 117605, 234773, 274761, 407469, 553681, 792613, 1052525, 1493033, 1959009, 3135537, 3904129, 5475673, 7173725, 9853325

Offset: 0

Author

Gus Wiseman, Nov 15 2018

Keywords

• nonn

Examples

			The a(5) = 9 matrices:
  [5] [4 1] [3 2] [2 3] [1 4]
.
  [4] [3] [2] [1]
  [1] [2] [3] [4]
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000005, A000219, A007716, A008289, A114736, A117433, A120733, A321645, A321653, A321655, A321659, A321661, A321662.

Programs

Mathematica

prs2mat[prs_]:=Table[Count[prs,{i,j}],{i,Union[First/@prs]},{j,Union[Last/@prs]}];
multsubs[set_,k_]:=If[k==0,{{}},Join@@Table[Prepend[#,set[[i]]]&/@multsubs[Drop[set,i-1],k-1],{i,Length[set]}]];
Table[Length[Select[multsubs[Tuples[Range[n],2],n],And[Union[First/@#]==Range[Max@@First/@#],Union[Last/@#]==Range[Max@@Last/@#],UnsameQ@@Join@@prs2mat[#]]&]],{n,5}]

PARI

seq(n)={my(B=vector((sqrtint(8*(n+1))+1)\2, n, if(n==1, 1, (n-1)!*numdiv(n-1) + n!*(numdiv(n) - 2)))); apply(p->sum(i=0, poldegree(p), B[i+1]*polcoef(p, i)), Vec(prod(k=1, n, 1 + x^k*y + O(x*x^n))))} \\ Andrew Howroyd, Nov 16 2018

Formula

a(n) = Sum_{k>=1} (k!*A000005(k) + (k+1)!*(A000005(k+1) - 2))*A008289(n,k) for n > 0. - Andrew Howroyd, Nov 17 2018

Extensions

Terms a(11) and beyond from Andrew Howroyd, Nov 16 2018