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A160085 Number of ordered complementing systems of subsets of {0, 1, ..., n-1} (see A104725). 0
0, 1, 1, 1, 3, 1, 5, 1, 13, 3, 5, 1, 33, 1, 5, 5, 75, 1, 33, 1, 33, 5, 5, 1, 261, 3, 5, 13, 33, 1, 61, 1, 541, 5, 5, 5, 375, 1, 5, 5, 261, 1, 61, 1, 33, 33, 5, 1, 2405, 3, 33, 5, 33, 1, 261, 5, 261, 5, 5, 1, 717, 1, 5, 33, 4683, 5, 61, 1, 33, 5, 61, 1, 4549, 1, 5, 33, 33, 5, 61, 1, 2405 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

Also number of permuted labeled factorizations of n (see the Munagi link for definition and examples)

LINKS

Table of n, a(n) for n=0..80.

A. O. Munagi, k-Complementing Subsets of Nonnegative Integers, International Journal of Mathematics and Mathematical Sciences, 2005:2 (2005), 215-224.

A. O. Munagi, Labeled factorization of integers, Electron. J. Combin. 16 (2009), no. 1, Research Paper 50, 17pp.

FORMULA

a(0) = 0, a(1) = 1, a(n) = Sum_{j=1..A001222(n)} (Sum_{k=1..j} k!Stirling2(j-1,k-1)), n > 1, where ordfac(n,k) = number of ordered factorizations of n into k factors.

EXAMPLE

a(6) = 5: there are 3 complementing systems of subsets of {0,1,2,3,4,5} namely {{0,1,2,3,4,5}}, {{0,1,2},{0,3}} and {{0,1},{0,2,4}} (see A104725). Permuting the components gives 2 additional systems: {{0,3},{0,1,2}} and {{0,2,4},{0,1}}. Thus since {{0,1,2},{0,3}} is a complementing system of subsets of {0,1,2,3,4,5} we have 0 = 0 + 0, 1 = 1 + 0, 2 = 2 + 0, 3 = 0 + 3, 4 = 1 + 3, 5 = 2 + 3.

MAPLE

a:=proc(n::integer) local u, r, i, j, k; if n<1 then return 0; elif n=1 then return 1; end if; u:=map(x->x[2], ifactors(n)[2]); r:=add(u[i], i=1..nops(u)); add(add(k!*add((-1)^i*binomial(t, i)*product(binomial(u[j]+t-i-1, u[j]), j=1..nops(u)), i=0..t-1)*stirling2(t-1, k-1), t=k..r), k=1..r); end proc: seq(a(n), n=0..99);

CROSSREFS

Cf. A104725.

Sequence in context: A146059 A046531 A289310 * A147024 A146378 A166086

Adjacent sequences:  A160082 A160083 A160084 * A160086 A160087 A160088

KEYWORD

nonn

AUTHOR

Augustine O. Munagi, May 01 2009

STATUS

approved

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Last modified April 22 11:46 EDT 2019. Contains 322330 sequences. (Running on oeis4.)