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%I #21 Dec 17 2023 10:31:29
%S 1,1,1,1,1,1,3,1,6,1,10,3,1,15,14,1,21,40,1,1,28,90,16,1,36,175,85,1,
%T 45,308,295,9,1,55,504,805,105,1,66,780,1876,594,2,1,78,1155,3906,
%U 2331,76,1,91,1650,7470,7280,771,1,105,2288,13365,19404,4529,30,1,120,3094,22660,45990,19348,650
%N Triangle read by rows: T(n,k) is the number of secondary structures of size n with no stacks of length >=2 and having k stacks of length 1 (n>=0, k>=0).
%C For "secondary structure" and "stack" see the Hofacker et al. reference, p. 209.
%C Sum of entries in row n is A202850(n).
%C From _Sergey Kitaev_, Nov 18 2023: (Start)
%C Number of permutations of length n avoiding simultaneously the patterns 132 and 321 with k descents (k>=0). A descent in a permutation a(1)a(2)...a(n) is position i such that a(i)>a(i+1).
%C Number of permutations of length n avoiding simultaneously the patterns 132 and 321 with k left-to-right minima (k>=1). A left-to-right minimum in a permutation a(1)a(2)...a(n) is position i such that a(j)>a(i) for all j<i. (End)
%H Tian Han and Sergey Kitaev, <a href="https://arxiv.org/abs/2311.02974">Joint distributions of statistics over permutations avoiding two patterns of length 3</a>, arXiv:2311.02974 [math.CO], 2023.
%H I. L. Hofacker, P. Schuster, and P. F. Stadler, <a href="https://doi.org/10.1016/S0166-218X(98)00073-0">Combinatorics of RNA secondary structures</a>, Discrete Appl. Math., 88, 1998, 207-237.
%H P. R. Stein and M. S. Waterman, <a href="https://doi.org/10.1016/0012-365X(79)90033-5">On some new sequences generalizing the Catalan and Motzkin numbers</a>, Discrete Math., 26 (1979), 261-272.
%F G.f. G(t,z) satisfies G = 1 + zG + [tz^2/(1 + tz^2)]G(G-1).
%F The multivariate g.f. H(z, t[1], t[2], ...) of secondary structures with respect to size (marked by z) and number of stacks of length j (marked by t[j]) satisfies H = 1 + zH + (f/(1 + f))H(H-1), where f = t[1]z^2 + t[2]z^4 + t[3]z^6 + ... .
%F From _Sergey Kitaev_, Nov 18 2023: (Start)
%F G.f.: (1 - 2*x + x^2 + q*x^2)/(1 - x)^3, where x records length and q records descents.
%F G.f.: (1-3*x+s*x+3*x^2-2*s*x^2+s^2*x^2-x^3+s*x^3)/(1-x)^3, where x records length and s records left-to-right minima. (End)
%e Row 5 is 1,6: representing unpaired vertices by v and arcs by AA, BB, etc., the 7 (= A202850(5)) secondary structures of size 5 and with no stacks of length >=2 are vvvvv, AvAvv, vvAvA, AvvAv, vAvvA, AvvvA, vAvAv; with the exception of the first one, they all have 1 stack of length 1.
%e Triangle starts:
%e 1;
%e 1;
%e 1;
%e 1,1;
%e 1,3;
%e 1,6;
%e 1,10,3;
%e 1,15,14;
%p eq := G = 1+z*G+t*z^2*G*(G-1)/(1+t*z^2): G := RootOf(eq, G): Gser := simplify(series(G, z = 0, 22)): for n from 0 to 17 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 17 do seq(coeff(P[n], t, k), k = 0 .. degree(P[n])) end do; # yields sequence in triangular form
%Y Cf. A202850
%K nonn,tabf
%O 0,7
%A _Emeric Deutsch_, Dec 27 2011
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