A112354 - OEIS

%I #35 Nov 21 2022 08:22:13

%S 1,1,4,17,92,572,4156,34159,314368,3199844,35703996,433421495,

%T 5687955724,80256874912,1211781887796,19496946534720,333041104402860,

%U 6019770246910128,114794574818830716,2303332661416242633,48509766592884311132,1069983257387132347080

%N Inverse Euler transform of n!. Also the number of sequences of permutations with no global descents which are Lyndon (smallest in lexicographic order of all cyclic shifts of the sequences) where the size of the sequence = sum of sizes of the permutations.

%H Seiichi Manyama, <a href="/A112354/b112354.txt">Table of n, a(n) for n = 1..449</a> (terms 1..200 from Alois P. Heinz)

%H M. Aguiar and A. Lauve, <a href="https://pdfs.semanticscholar.org/a511/fc8121e916907c046730a803d6725549ef4b.pdf">Antipode and Convolution Powers of the Identity in Graded Connected Hopf Algebras</a>, FPSAC 2013 Paris, France DMTCS Proc. AS, 2013, 1083-1094.

%H M. Aguiar, A. Lauve, <a href="https://arxiv.org/abs/1403.7584">The characteristic polynomial of the Adams operators on graded connected Hopf algebras</a>, arXiv:1403.7584 [math.RA], 2014.

%F Product_{k>=1} 1/(1-x^k)^{a(k)} = Sum_{n>=0} n! x^n.

%F a(n) ~ n! * (1 - 1/n - 1/n^2 - 4/n^3 - 23/n^4 - 171/n^5 - 1542/n^6 - 16241/n^7 - 194973/n^8 - 2622610/n^9 - 39027573/n^10 - ...), for coefficients see A113869. - _Vaclav Kotesovec_, Sep 04 2014, extended Nov 27 2020

%e a(3) = 4 because (123), (213), (132) and (1,21) are all Lyndon.

%e a(4) = 17 because there are 13 permutations with no global descents of size 4 and (1,123), (1,213), (1,132) are all Lyndon.

%e a(5) = 92 = 71 permutations with no global descents+13 sequences of the form (1,pi) where pi in S_4 with no global descents+(1,1,1,21),(1,21,21),(1,1,123),(1,1,213),(1,1,132),(21,123),(21,213),(21,132).

%p read transforms; EULERi([seq(n!,n=1..30)]);

%p # The function EulerInvTransform is defined in A358451.

%p a := EulerInvTransform(factorial):

%p seq(a(n), n = 1..22); # _Peter Luschny_, Nov 21 2022

%t ff = Range[n = 22]!; s = {}; For[i = 1, i <= n, i++, AppendTo[s, i*ff[[i]] - Sum[s[[d]]*ff[[i-d]], {d, i-1}]]]; Table[Sum[If[Divisible[i, d], MoebiusMu[i/d], 0]*s[[d]], {d, 1, i}]/i, {i, n}] (* _Jean-François Alcover_, Apr 15 2016 *)

%Y Cf. A003319, A000142, A305786.

%K nonn

%O 1,3

%A _Mike Zabrocki_, Sep 05 2005