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

1,2

Links

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

Formula

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

Example

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]

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 *)

Prog

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

Crossrefs

Cf. A000005, A000670, A060223, A101509.

Cf. A323351, A323869, A323870, A323871.

Sequence in context: A211946 A027283 A009639 * A226980 A199591 A230867

Adjacent sequences: A323865 A323866 A323867 * A323869 A323870 A323871

Keyword

nonn

Author

Gus Wiseman, Feb 04 2019

Status

approved