OFFSET
1,2
COMMENTS
A set composition of n is an ordered sequence [S_1, S_2, ..., S_k] where S_i subset of [n] all disjoint and the union of all S_i is [n] (see A000670). A set composition is atomic if S_1 union ... union S_j does not equal [r] for any r < n and j < k. a(n) is the number of atomic set compositions.
A preference function of n is a word of length n where all the numbers 1 through k occur at least once for some k <= n (see A000670). A preference function is atomic if no strict leading subword contains the only occurrences in the word of the letters 1 through j < k. a(n) is the number of atomic preference functions.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..400
Hugo Mlodecki, Decompositions of packed words and self duality of Word Quasisymmetric Functions, arXiv:2205.13949 [math.CO], 2022. See Table 2 p. 8.
FORMULA
G.f.: 1 - 1/Sum_{k>=0} A000670(k)*q^k.
G.f.: x/(1-2x/(1-2x/(1-4x/(1-3x/(1-6x/(1-4x/(1-8x/(1-5x/(1-...(continued fraction). - Philippe Deléham, Nov 22 2011
G.f.: (1-T(0))/x, where T(k) = 1 - x*(k+1)/(1 - 2*x*(k+1)/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 29 2013
Let A(x) be the g.f. A095989, B(x) the g.f. A000670, then A(x) = (1 - 1/B(x))/x. - Sergei N. Gladkovskii, Nov 29 2013
a(n) ~ n! / (2 * log(2)^(n+1)). - Vaclav Kotesovec, Oct 09 2019
EXAMPLE
Atomic set compositions a(1)=1: [{1}]; a(2)=2: [{12}], [{2},{1}]; a(3)=8: [{123}], [{2},{13}], [{3}, {12}], [{23}, {1}], [{13},{2}], [{2},{3},{1}], [{3},{1},{2}], [{3},{2},{1}].
Atomic preference functions a(1) = 1: 1; a(2)=2: 11, 21; a(3)=8: 111, 212, 221, 211, 121, 312, 231, 321.
MAPLE
MATHEMATICA
max = 20; Fubini[n_, r_] := Sum[k!*Sum[(-1)^(i+k+r)*(i+r)^(n-r)/(i!*(k-i-r)!), {i, 0, k-r}], {k, r, n}]; Fubini[0, 1] = 1; s = 1 - 1/Sum[ Fubini[k, 1] q^k, {k, 0, max}] + O[q]^max; CoefficientList[s/q, q] (* Jean-François Alcover, Mar 31 2016 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Mike Zabrocki, Jul 18 2004
STATUS
approved