1,2

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.

G.f.: 1-1/sum( A000670(k)*q^k, k >= 0)

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

A000670 := proc(n) option remember; local k; if n <=1 then 1 else add(binomial(n, k)*A000670(n-k), k=1..n); fi; end: add(A000670(k)*x^k, k=0..20): series(1-1/%, x, 21): [seq(coeff(%, x, i), i=1..20)];

Cf. A000670, A074664, A095993.

Sequence in context: A085615 A054726 A003576 this_sequence A124453 A000165 A109664

Adjacent sequences: A095986 A095987 A095988 this_sequence A095990 A095991 A095992

nonn

Mike Zabrocki (zabrocki(AT)mathstat.yorku.ca), Jul 18 2004