A202851 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).

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, 45, 308, 295, 9, 1, 55, 504, 805, 105, 1, 66, 780, 1876, 594, 2, 1, 78, 1155, 3906, 2331, 76, 1, 91, 1650, 7470, 7280, 771, 1, 105, 2288, 13365, 19404, 4529, 30, 1, 120, 3094, 22660, 45990, 19348, 650

Offset

0,7

Comments

For "secondary structure" and "stack" see the Hofacker et al. reference, p. 209.

Sum of entries in row n is A202850(n).

From Sergey Kitaev, Nov 18 2023: (Start)

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).

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)

Links

Table of n, a(n) for n=0..68.

Tian Han and Sergey Kitaev, Joint distributions of statistics over permutations avoiding two patterns of length 3, arXiv:2311.02974 [math.CO], 2023.

I. L. Hofacker, P. Schuster, and P. F. Stadler, Combinatorics of RNA secondary structures, Discrete Appl. Math., 88, 1998, 207-237.

P. R. Stein and M. S. Waterman, On some new sequences generalizing the Catalan and Motzkin numbers, Discrete Math., 26 (1979), 261-272.

Formula

G.f. G(t,z) satisfies G = 1 + zG + [tz^2/(1 + tz^2)]G(G-1).

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 + ... .

From Sergey Kitaev, Nov 18 2023: (Start)

G.f.: (1 - 2*x + x^2 + q*x^2)/(1 - x)^3, where x records length and q records descents.

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)

Example

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.
Triangle starts:
1;
1;
1;
1,1;
1,3;
1,6;
1,10,3;
1,15,14;

Maple

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

Crossrefs

Cf. A202850

Sequence in context: A304236 A360654 A145063 * A007650 A236540 A339384

Adjacent sequences: A202848 A202849 A202850 * A202852 A202853 A202854

Keyword

nonn,tabf

Author

Emeric Deutsch, Dec 27 2011

Status

approved