A321659 Number of nonnegative integer matrices with sum of entries equal to n and no zero rows or columns, whose nonzero entries are all distinct.

1, 1, 1, 9, 9, 17, 161, 169, 313, 465, 5313, 5465, 10457, 15313, 25009, 271929, 286329, 537953, 799121, 1297369, 1805161, 20532897, 21292017, 40508297, 59738825, 97431073, 135137569, 209525865, 2089381929, 2200470833, 4135252289, 6124698121, 9937836505

Offset

0,4

Links

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

Formula

a(n) = Sum_{k>=1} A101370(k)*k!*A008289(n,k) for n > 0. - Andrew Howroyd, Nov 17 2018

Example

The a(5) = 17 matrices:
  [5] [4 1] [3 2] [2 3] [1 4]
.
  [4] [4 0] [3] [3 0] [2] [2 0] [1] [1 0] [0 4] [0 3] [0 2] [0 1]
  [1] [0 1] [2] [0 2] [3] [0 3] [4] [0 4] [1 0] [2 0] [3 0] [4 0]

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@@DeleteCases[Join@@prs2mat[#], 0]]&]], {n, 5}]

Prog

(PARI) \\ here b(n) is A101370(n).
b(n)={sum(m=0, n, sum(k=0, m, stirling(m, k, 2)*k!)^2*polcoef(log(1+x+O(x*x^n))^m, n)/m!)}
seq(n)={my(B=vector((sqrtint(8*(n+1))+1)\2, n, b(n-1))); apply(p->sum(i=0, poldegree(p), B[i+1]*i!*polcoef(p, i)), Vec(prod(k=1, n, 1 + x^k*y + O(x*x^n))))} \\ Andrew Howroyd, Nov 16 2018

Crossrefs

Cf. A000219, A007716, A008289, A101370, A114736, A117433, A120733, A321645, A321653, A321655, A321660, A321661, A321662.

Sequence in context: A344335 A168390 A333151 * A040073 A003886 A065999

Adjacent sequences: A321656 A321657 A321658 * A321660 A321661 A321662

Keyword

nonn

Author

Gus Wiseman, Nov 15 2018

Extensions

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

Status

approved