A178841 The number of pure inverting compositions of n.

1, 0, 0, 1, 2, 5, 9, 19, 37, 74, 148, 296, 591, 1183, 2366, 4731, 9463, 18926, 37852, 75704, 151408, 302816, 605633, 1211265, 2422530, 4845060, 9690121, 19380241, 38760482, 77520964, 155041928, 310083856, 620167712, 1240335424, 2480670848, 4961341695, 9922683391, 19845366782, 39690733564

Offset: 0

Author

Aaron Lauve (lauve(AT)math.luc.edu), Jun 17 2010

Keywords

• nonn

Comments

A. Garsia and N. Wallach show that the algebra of quasisymmetric functions is a free module over the algebra of symmetric functions.

The pure inverting compositions index a basis for this module, as conjectured by F. Bergeron and C. Reutenauer.

Georg Fischer observes that the terms of this sequence are very similar to those of A152537. This may be just a coincidence, caused by the fact that their generating functions are almost identical. - N. J. A. Sloane, Mar 23 2019

Examples

			Call a composition w=w1w2...wk "inverting" if for all N > 1 appearing within the word w, there is a pair i < j with w_i = N and w_j = N-1. Factor a composition w as w=uv, with v of maximal length taking the form k^d ... 3^c 2^b 1^a. Call w "pure" if k is even.
Let A(n) be the pure inverting compositions of n, so that a(n) = #A(n). For example, A(3) = {21}, A(4) = {121, 211}, A(5) = {212, 221, 1121, 1211, 2111}.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Anders Claesson, Atli Fannar Franklín, and Einar Steingrímsson, Permutations with few inversions, arXiv:2305.09457 [math.CO], 2023.
• A. Garsia and N. Wallach, Qsym over Sym is free, J. Combin. Theory Ser. A 104 (2003), no. 2, 217--263.
• A. Lauve and S. Mason, Qsym over Sym has a stable basis, arXiv:1003.2124 [math.CO], 2010.
• Eric Weisstein's World of Mathematics, q-Factorial.

Crossrefs

Cf. A000041, A011782, A152537.

Programs

Magma

m:=45; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( ((1-q)/(1-2*q))*(&*[1-q^k: k in [1..m]]) )); // G. C. Greubel, Jan 21 2019

Mathematica

With[{m = 45}, CoefficientList[Series[((1-q)/(1-2*q))*Product[(1-q^k), {k, 1, m+2}], {q, 0, m}], q]] (* G. C. Greubel, Jan 21 2019 *)

PARI

m=45; my(q='q+O('q^m)); Vec(((1-q)/(1-2*q))*prod(k=1,m+2,(1-q^k))) \\ G. C. Greubel, Jan 21 2019

Sage

m=45; (((1-x)/(1-2*x))*prod(1-x^k for k in (1..m))).series(x, m).coefficients(x, sparse=False) # G. C. Greubel, Jan 21 2019

Formula

G.f.: P(q) = ((1-q)/(1-2*q))*(Product_{k>=1} (1-q^k)) = 1 + Sum_{n>=1} a(n)*q^n = the g.f. for A011782 divided by the g.f. for A000041.

Define P(m,q) recursively by P(0,q) = 1; P(m,q) = P(m-1,q) + q^m*(m!q - P(m-1,q)). (Here m!_q is the standard q-factorial.) Then P(m,q) enumerates the pure inverting compositions of length at most m and lim{m->infinity} P(m,q) = P(q).

Define a(n,0) = a(n); a(n,1) = a(0) + ... + a(n); and a(n,k) = a(n,k-1) + a(n-k,k+1) + a(n-2k, n+1) + ... Then a(n) + a(n-1,1) + a(n-2,2) + ... + a(0,n) = A011782(n), the number of compositions of n. - Gregory L. Simay, Jun 03 2019

The convolution of a(n) with A000041(n), the partitions of n, is A011782(n). - Gregory L. Simay, Jun 03 2019

Extensions

Terms a(26) onward added by G. C. Greubel, Jan 21 2019