A323868 Number of matrices of size n whose entries cover an initial interval of positive integers.

1, 6, 26, 225, 1082, 18732, 94586, 2183340, 21261783, 408990252, 3245265146, 168549405570, 1053716696762, 42565371881772, 921132763911412, 26578273409906775, 260741534058271802, 20313207979541071938, 185603174638656822266, 16066126777466305218690

Offset: 1

Author

Gus Wiseman, Feb 04 2019

Keywords

• nonn

Examples

			The 42 matrices of size 4 whose entries cover {1,2}:
  1222 2111 1122 2211 1212 2121 1221 2112 1112 2221 1121 2212 1211 2122
.
  12  21  11  22  12  21  12  21  11  22  11  22  12  21
  22  11  22  11  12  21  21  12  12  21  21  12  11  22
.
  1   2   1   2   1   2   1   2   1   2   1   2   1   2
  2   1   1   2   2   1   2   1   1   2   1   2   2   1
  2   1   2   1   1   2   2   1   1   2   2   1   1   2
  2   1   2   1   2   1   1   2   2   1   1   2   1   2
The 18 matrices of size 4 whose entries cover {1,2} with multiplicities {2,2}:
  [1 1 2 2] [2 2 1 1] [1 2 1 2] [2 1 2 1] [1 2 2 1] [2 1 1 2]
.
  [1 1] [2 2] [1 2] [2 1] [1 2] [2 1]
  [2 2] [1 1] [1 2] [2 1] [2 1] [1 2]
.
  [1] [2] [1] [2] [1] [2]
  [1] [2] [2] [1] [2] [1]
  [2] [1] [1] [2] [2] [1]
  [2] [1] [2] [1] [1] [2]
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..424

Crossrefs

Cf. A000005, A000670, A060223, A101509.

Cf. A323351, A323869, A323870, A323871.

Programs

Maple

b:= proc(n) option remember; `if`(n=0, 1,
      add(b(n-j)*binomial(n, j), j=1..n))
    end:
a:= n-> b(n)*numtheory[tau](n):
seq(a(n), n=1..20);  # Alois P. Heinz, Feb 04 2019

Mathematica

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
nrmmats[n_]:=Join@@Table[Table[Table[Position[stn,{i,j}][[1,1]],{i,d},{j,n/d}],{stn,Join@@Permutations/@sps[Tuples[{Range[d],Range[n/d]}]]}],{d,Divisors[n]}];
Table[Length[nrmmats[n]],{n,6}]
Table[DivisorSigma[0, n]*Sum[k! StirlingS2[n, k], {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Feb 05 2019 *)

PARI

a(n) = numdiv(n)*sum(k=0, n, stirling(n, k, 2)*k!); \\ Michel Marcus, Feb 05 2019

Formula

a(n) = A000005(n) * A000670(n).