%I #28 May 30 2022 02:25:34

%S 1,2,8,48,368,3376,35824,430512,5773936,85482032,1384936688,

%T 24380214960,463522810736,9468048895792,206831329017328,

%U 4812581925690288,118843801816575088,3104590192664327216,85544737118902122224,2479681575659312797872,75434373300016828382576

%N INVERTi transform applied to the ordered Bell numbers.

%C 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.

%C 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.

%H Alois P. Heinz, <a href="/A095989/b095989.txt">Table of n, a(n) for n = 1..400</a>

%H Hugo Mlodecki, <a href="https://arxiv.org/abs/2205.13949">Decompositions of packed words and self duality of Word Quasisymmetric Functions</a>, arXiv:2205.13949 [math.CO], 2022. See Table 2 p. 8.

%F G.f.: 1 - 1/Sum_{k>=0} A000670(k)*q^k.

%F 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

%F 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

%F 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

%F a(n) ~ n! / (2 * log(2)^(n+1)). - _Vaclav Kotesovec_, Oct 09 2019

%e 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}].

%e Atomic preference functions a(1) = 1: 1; a(2)=2: 11, 21; a(3)=8: 111, 212, 221, 211, 121, 312, 231, 321.

%p 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)];

%t 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 *)

%Y Cf. A000670, A074664, A095993.

%K nonn

%O 1,2

%A _Mike Zabrocki_, Jul 18 2004