OFFSET
0,3
COMMENTS
Number of pushall stack words of length 3n. These consist of n 'ρ' letters, n 'λ' letters and n 'μ' letters, such that the count of 'λ' letters never exceeds the count of 'ρ' letters, the count of 'μ' letters never exceeds the count of 'λ' letters, and all the 'ρ' letters occur before any of the 'μ' letters.
A permutation of length n is 2-stack pushall sortable if and only if it can be sorted by a sequence of 3n operations represented by a pushall stack word of length 3n, where ρ corresponds to pushing to the 1st (Right) stack, λ corresponds to popping from the 1st stack and pushing to the 2nd (Left) stack, and μ corresponds to popping from the 2nd stack.
There is an obvious bijection between pushall stack words of length 3n using the letters 'ρ', 'λ', and 'μ', and pushall stack words of length 3n in which 'ρ' and 'μ' are the same symbol. In this way, a(n) is the number of words consisting of n 'λ' letters and 2n 'μ' letters, such that the count of 'λ' letters never exceeds the count of 'μ' letters in any prefix or suffix of the word. This allows a closed form (added below) based on two usages of "Andre's reflection method", in analogy with the Catalan numbers. - Janis Stipins, May 27 2019
LINKS
David Bevan, Table of n, a(n) for n = 0..300
Nicolas Borie and Justine Falque, Product-Coproduct Prographs and Triangulations of the Sphere, arXiv:2202.05757 [math.CO], 2022. Proceedings of the 34th Conference on Formal Power Series and Algebraic Combinatorics (Bangalore), Séminaire Lotharingien de Combinatoire 86B (2022) Article #82. See Proposition 1, p. 10. See also.
Ryota Inagaki and Dimana Pramatarova, On Semisymmetric Height and a Multidimensional Generalization of Weighted Catalan Numbers, arXiv:2604.04900 [math.CO], 2026. See p. 29.
Adeline Pierrot and Dominique Rossin, 2-stack pushall sortable permutations, arXiv:1303.4376 [cs.DM], 2013.
FORMULA
The o.g.f. f=f(z) is algebraic, satisfying the cubic equation (1-16*z+64*z^2) + (-1+21*z-96*z^2)*f + (-4*z+27*z^2)*f^2 + (-4*z^2+27*z^3)*f^3 = 0.
a(n) = A259475(n,n). - Alois P. Heinz, Nov 19 2018
a(n) = binomial(3*n,n) - 2*binomial(3*n,n-1) + binomial(3*n,n-2). - Janis Stipins, May 27 2019
G.f.: (2*(1 - 6*x)*cos(arccos(1 - (27*x)/2)/6)/sqrt(4 - 27*x) + 4*sqrt(3)*sqrt(x)*sin(arcsin(3*sqrt(3)*sqrt(x)/2)/3) - 1)/(3*x). - Stefano Spezia, Feb 19 2022
a(n) ~ 3^(3*n+1/2) / (2^(2*n+3) * sqrt(Pi*n)). - Amiram Eldar, Dec 07 2025
EXAMPLE
For n=2, the four walks are EEDDSS, EEDSDS, EDEDSS and EDESDS.
MATHEMATICA
CoefficientList[Module[{r=0}, Do[r+=Coefficient[1-16z+64z^2+(21z-96z^2)f+(-4z+27z^2)f^2+(-4z^2+27z^3)f^3/.f->r, z, i]z^i, {i, 0, 30}]; r], z]
PROG
(PARI) N=O(z^35); f=1+N; while(f+N<>f=1+(5*z-32*z^2+(-4+27*z)*z*(1+z*f)*f^2)/(1-21*z+96*z^2), ); Vec(f+N) \\ Using that the g.f. is fixed point of T(f)=1+(5*z-32*z^2+(-4+27*z)*z*(1+z*f)*f^2)/(1-21*z+96*z^2). - M. F. Hasler, Jul 13 2016
(PARI) a(n) = binomial(3*n, n) - 2*binomial(3*n, n-1) + binomial(3*n, n-2); \\ Janis Stipins, May 27 2019
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
David Bevan, Jul 13 2016
EXTENSIONS
Data double-checked by M. F. Hasler, Jul 13 2016
STATUS
approved