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%I #44 Mar 22 2022 02:25:18
%S 1,1,4,21,121,728,4488,28101,177859,1134705,7283640,46981740,
%T 304253964,1976886616,12880883408,84130964709,550649378199,
%U 3610705776755,23714554702020,155979407872365,1027269675638745,6773476758296220,44709685668953760,295402076512228140,1953492865541875476
%N Number of walks in the first quadrant starting and ending at (0,0) consisting of 3n steps taken from {E=(1, 0), D=(-1, 1), S=(0, -1)}, no S step occurring before the final E step.
%C 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.
%C 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.
%C 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
%H David Bevan, <a href="/A274969/b274969.txt">Table of n, a(n) for n = 0..300</a>
%H Nicolas Borie and Justine Falque, <a href="https://arxiv.org/abs/2202.05757">Product-Coproduct Prographs and Triangulations of the Sphere</a>, arXiv:2202.05757 [math.CO], 2022. Proceedings of the 34th Conference on Formal Power Series and Algebraic Combinatorics (Bangalore), Séminaire Lotharingien de Combinatoire XX (2022). See Proposition 1, p. 10.
%H Adeline Pierrot and Dominique Rossin, <a href="http://arxiv.org/abs/1303.4376">2-stack pushall sortable permutations</a>, arXiv:1303.4376 [cs.DM], 2013.
%F 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.
%F a(n) = A259475(n,n). - _Alois P. Heinz_, Nov 19 2018
%F a(n) = binomial(3*n,n) - 2*binomial(3*n,n-1) + binomial(3*n,n-2). - _Janis Stipins_, May 27 2019
%F 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
%e For n=2, the four walks are EEDDSS, EEDSDS, EDEDSS and EDESDS.
%t 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]
%o (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
%o (PARI) a(n) = binomial(3*n,n) - 2*binomial(3*n,n-1) + binomial(3*n,n-2); \\ _Janis Stipins_, May 27 2019
%Y Walks in the first quadrant from (0,0) to (0,0) with steps from {E, D, S} A005789.
%Y 2-stack pushall sortable permutations A274970.
%Y Cf. A259475.
%K nonn,easy
%O 0,3
%A _David Bevan_, Jul 13 2016
%E Data double-checked by _M. F. Hasler_, Jul 13 2016
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