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Dimensions of primitive Lie algebras connected with the Mantaci-Reutenauer algebra MR^(3).
3

%I #61 Jun 19 2019 08:16:30

%S 3,6,20,60,204,670,2340,8160,29120,104754,381300,1397740,5162220,

%T 19172790,71582716,268431360,1010580540,3817733920,14467258260,

%U 54975528948,209430785460,799644629550,3059510616420,11728123327840,45035996273664,173215367702370

%N Dimensions of primitive Lie algebras connected with the Mantaci-Reutenauer algebra MR^(3).

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

%C From _Petros Hadjicostas_, Jun 18 2019: (Start)

%C 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.

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

%C It is also the CHK (circular, identity, unlabeled) transform of the sequence 3, 3, 3, ... See the link by Bowers about such transforms.

%C 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) = 3 for all n >= 1, and thus, C(x) = 3*x/(1 - x). It follows that the g.f. of the output sequence is -Sum_{d >= 1} (mu(d)/d) * log(1 - 3*x^d/(1 - x^d)).

%C (End)

%H G. C. Greubel, <a href="/A185172/b185172.txt">Table of n, a(n) for n = 1..1650</a>

%H C. G. Bower, <a href="/transforms2.html">Transforms (2)</a>.

%H Jean-Christophe Novelli and Jean-Yves Thibon, <a href="http://arxiv.org/abs/0806.3682">Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions</a>, arXiv:0806.3682 [math.CO], 2008. See Eqs. (94) and (95).

%H Jean-Christophe Novelli and Jean-Yves Thibon, <a href="https://doi.org/10.1016/j.disc.2010.09.008">Free quasi-symmetric functions and descent algebras for wreath products, and noncommutative multi-symmetric functions</a>, Discrete Math. 310 (2010), no. 24, 3584-3606. See Eqs. (99) and (100).

%F From _Petros Hadjicostas_, Jun 17 2019: (Start)

%F a(1) = 3 and a(n) = (1/n) * Sum_{d|n} mu(d) * 4^(n/d) for n > 1 (from Eq. (95) in Novelli and Thibon (2008) or Eq. (100) in Novelli and Thibon (2010)).

%F a(n) = (1/n) * Sum_{d|n} mu(d) * (4^(n/d) - 1) = (1/n) * Sum_{d|n} mu(d) *A024036(n/d) for n >= 1.

%F G.f.: -Sum_{d >= 1} (mu(d)/d) * log(1 - 3*x^d/(1 - x^d)) = -x - Sum_{d >= 1} (mu(d)/d) * log(1 - 4*x^d).

%F (End)

%p From _Petros Hadjicostas_, Jun 18 2019: (Start)

%p Suppose we have three colors, say, A, B, C. Here, a(1) = 3 because we have the following aperiodic unmarked cyclic compositions of n = 1: 1_A, 1_B, 1_C.

%p We have a(2) = 6 because we have the following aperiodic unmarked cyclic compositions of n = 2: 2_A, 2_B, 2_C, 1_A + 1_B, 1_B + 1_C, 1_C + 1_A.

%p We have a(3) = 20 because we have the following aperiodic unmarked cyclic compositions of n = 3: 3_X, where X \in {A, B, C}; 1_X + 2_Y, where (X, Y) \in {(A, A), (A, B), (A, C), (B, A), (B, B), (B, C), (C, A), (C, B), (C, C)}; 1_A + 1_B + 1_C and 1_C + 1_B + 1_A; and 1_X + 1_Y + 1_Y, where (X, Y) \in {(A, B), (A, C), (B, A), (B, C), (C, A), (C, B)}.

%p (End)

%t a[1] = 3; a[n_] := DivisorSum[n, MoebiusMu[#]*4^(n/#)&]/n; Array[a, 26] (* _Jean-François Alcover_, Dec 07 2015, adapted from PARI *)

%o (PARI) a(l=3,n) = if (n==1, l, sumdiv(n, d, moebius(d)*(1+l)^(n/d))/n); \\ _Michel Marcus_, Feb 09 2013

%Y Cf. A024036, A027377, A032253, A185171.

%K nonn

%O 1,1

%A _N. J. A. Sloane_, Jan 23 2012

%E More terms from _Michel Marcus_, Feb 09 2013

%E Name edited by _Petros Hadjicostas_, Jun 17 2019