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

%I A323868 #13 Feb 06 2019 09:19:33
%S A323868 1,6,26,225,1082,18732,94586,2183340,21261783,408990252,3245265146,
%T A323868 168549405570,1053716696762,42565371881772,921132763911412,
%U A323868 26578273409906775,260741534058271802,20313207979541071938,185603174638656822266,16066126777466305218690
%N A323868 Number of matrices of size n whose entries cover an initial interval of positive integers.
%H A323868 Alois P. Heinz, <a href="/A323868/b323868.txt">Table of n, a(n) for n = 1..424</a>
%F A323868 a(n) = A000005(n) * A000670(n).
%e A323868 The 42 matrices of size 4 whose entries cover {1,2}:
%e A323868   1222 2111 1122 2211 1212 2121 1221 2112 1112 2221 1121 2212 1211 2122
%e A323868 .
%e A323868   12  21  11  22  12  21  12  21  11  22  11  22  12  21
%e A323868   22  11  22  11  12  21  21  12  12  21  21  12  11  22
%e A323868 .
%e A323868   1   2   1   2   1   2   1   2   1   2   1   2   1   2
%e A323868   2   1   1   2   2   1   2   1   1   2   1   2   2   1
%e A323868   2   1   2   1   1   2   2   1   1   2   2   1   1   2
%e A323868   2   1   2   1   2   1   1   2   2   1   1   2   1   2
%e A323868 The 18 matrices of size 4 whose entries cover {1,2} with multiplicities {2,2}:
%e A323868   [1 1 2 2] [2 2 1 1] [1 2 1 2] [2 1 2 1] [1 2 2 1] [2 1 1 2]
%e A323868 .
%e A323868   [1 1] [2 2] [1 2] [2 1] [1 2] [2 1]
%e A323868   [2 2] [1 1] [1 2] [2 1] [2 1] [1 2]
%e A323868 .
%e A323868   [1] [2] [1] [2] [1] [2]
%e A323868   [1] [2] [2] [1] [2] [1]
%e A323868   [2] [1] [1] [2] [2] [1]
%e A323868   [2] [1] [2] [1] [1] [2]
%p A323868 b:= proc(n) option remember; `if`(n=0, 1,
%p A323868       add(b(n-j)*binomial(n, j), j=1..n))
%p A323868     end:
%p A323868 a:= n-> b(n)*numtheory[tau](n):
%p A323868 seq(a(n), n=1..20);  # _Alois P. Heinz_, Feb 04 2019
%t A323868 sps[{}]:={{}};sps[set:{i_,___}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,___}];
%t A323868 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]}];
%t A323868 Table[Length[nrmmats[n]],{n,6}]
%t A323868 Table[DivisorSigma[0, n]*Sum[k! StirlingS2[n, k], {k, 1, n}], {n, 1, 20}] (* _Vaclav Kotesovec_, Feb 05 2019 *)
%o A323868 (PARI) a(n) = numdiv(n)*sum(k=0, n, stirling(n, k, 2)*k!); \\ _Michel Marcus_, Feb 05 2019
%Y A323868 Cf. A000005, A000670, A060223, A101509.
%Y A323868 Cf. A323351, A323869, A323870, A323871.
%K A323868 nonn
%O A323868 1,2
%A A323868 _Gus Wiseman_, Feb 04 2019