A185171 Dimensions of primitive Lie algebras connected with the Mantaci-Reutenauer algebra MR^(2).

2, 3, 8, 18, 48, 116, 312, 810, 2184, 5880, 16104, 44220, 122640, 341484, 956576, 2690010, 7596480, 21522228, 61171656, 174336264, 498111952, 1426403748, 4093181688, 11767874940, 33891544368, 97764009000, 282429535752, 817028131140, 2366564736720

Offset

1,1

Comments

Maybe the definition should say: "Number of generators of degree n ...". The paper is a little unclear.

From Petros Hadjicostas, Jun 18 2019: (Start)

An unmarked cyclic composition of n >= 1 is an equivalence class of ordered partitions of n such that two such ordered partitions are equivalent iff one can be obtained from the other by rotation.

Here, a(n) is the number of aperiodic unmarked cyclic compositions of n where up to two colors can be used.

It is also the CHK (circular, identity, unlabeled) transform of the sequence 2, 2, 2, ... See the link by Bowers about such transforms.

If c = (c(n): n >= 1) is the input sequence with g.f. C(x) = Sum_{n >= 1} c(n)*x^n, then the g.f. of the output sequence ((CHK c)_d: d >= 1) is -Sum_{d >= 1} (mu(d)/d) * log(1 - C(x^d)). Here, c(n) = 2 for all n >= 1, and thus, C(x) = 2*x/(1 - x). It follows that the g.f. of the output sequence is -Sum_{d >= 1} (mu(d)/d) * log(1 - 2*x^d/(1 - x^d)).

(End)

Links

G. C. Greubel, Table of n, a(n) for n = 1..2000

C. G. Bower, Transforms (2).

Jean-Christophe Novelli and Jean-Yves Thibon, Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions, arXiv:0806.3682 [math.CO], 2008. See Eqs. (93) and (95).

Jean-Christophe Novelli and Jean-Yves Thibon, Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions, Discrete Math. 310 (2010), no. 24, 3584-3606. See Eqs. (98) and (100).

Formula

From Petros Hadjicostas, Jun 18 2019: (Start)

a(1) = 2 and a(n) = (1/n) * Sum_{d|n} mu(d) * 3^(n/d) for n > 1 (from Eq. (95) in Novelli and Thibon (2008) or Eq. (100) in Novelli and Thibon (2010)).

a(n) = (1/n) * Sum_{d|n} mu(d) * (3^(n/d) - 1) = (1/n) * Sum_{d|n} mu(d) * A024023(n/d) for n >= 1.

G.f.: -Sum_{d >= 1} (mu(d)/d) * log(1 - 2*x^d/(1 - x^d)) = -x - Sum_{d >= 1} (mu(d)/d) * log(1 - 3*x^d).

(End)

Example

From Petros Hadjicostas, Jun 18 2019: (Start)
Suppose we have two colors, say, A and B. Here, a(1) = 2 because we have the following aperiodic unmarked cyclic compositions of n = 1: 1_A and 1_B.
We have a(2) = 3 because we have the following aperiodic unmarked cyclic compositions of n = 2: 2_A, 2_B, and 1_A + 1_B.
We have a(3) = 8 because we have the following aperiodic unmarked cyclic compositions of n = 3: 3_A and 3_B; 1_X + 2_Y, where (X, Y) \in {(A, A), (A, B), (B, A), (B, B)}; 1_A + 1_B + 1_B and 1_B + 1_A + 1_A.
(End)

Mathematica

a[1] = 2; a[n_] := DivisorSum[n, MoebiusMu[#]*3^(n/#)&]/n; Array[a, 29] (* Jean-François Alcover, Dec 07 2015, adapted from PARI *)

Prog

(PARI) a(l=2, n) = if (n==1, l, sumdiv(n, d, moebius(d)*(1+l)^(n/d))/n); \\ Michel Marcus, Feb 09 2013

Crossrefs

Cf. A024023, A027376, A032251, A185172.

Sequence in context: A079224 A002369 A005957 * A339524 A158448 A073192

Adjacent sequences: A185168 A185169 A185170 * A185172 A185173 A185174

Keyword

nonn

Author

N. J. A. Sloane, Jan 23 2012

Extensions

More terms from Michel Marcus, Feb 09 2013

Name edited by Petros Hadjicostas, Jun 18 2019

Status

approved