OFFSET
0,5
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
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.
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
EXAMPLE
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}.
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 *)
PROG
(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
(Magma) m:=45; R<q>:=PowerSeriesRing(Integers(), m); Coefficients(R!( ((1-q)/(1-2*q))*(&*[1-q^k: k in [1..m]]) )); // 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
CROSSREFS
KEYWORD
nonn
AUTHOR
Aaron Lauve (lauve(AT)math.luc.edu), Jun 17 2010
EXTENSIONS
Terms a(26) onward added by G. C. Greubel, Jan 21 2019
STATUS
approved