OFFSET
1,2
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..424
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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Feb 04 2019
STATUS
approved