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A323868 Number of matrices of size n whose entries cover an initial interval of positive integers. 4
1, 6, 26, 225, 1082, 18732, 94586, 2183340, 21261783, 408990252, 3245265146, 168549405570, 1053716696762, 42565371881772, 921132763911412, 26578273409906775, 260741534058271802, 20313207979541071938, 185603174638656822266, 16066126777466305218690 (list; graph; refs; listen; history; text; internal format)
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

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Last modified July 25 16:03 EDT 2021. Contains 346291 sequences. (Running on oeis4.)