A113897 Triangle read by rows: number of simsun n-permutations with k descents.

1, 1, 1, 1, 4, 1, 11, 4, 1, 26, 34, 1, 57, 180, 34, 1, 120, 768, 496, 1, 247, 2904, 4288, 496, 1, 502, 10194, 28768, 11056, 1, 1013, 34096, 166042, 141584, 11056, 1, 2036, 110392, 868744, 1372088, 349504, 1, 4083, 349500, 4247720, 11204160, 6213288, 349504

Offset

1,5

Comments

Is this A094503 after removal of the top row? - R. J. Mathar, Aug 13 2008

Yes. See formula of Peter Bala, Jun 26 2012 in A094503. - Stefano Spezia, Aug 09 2023

Links

Table of n, a(n) for n=1..48.

Chak-On Chow and Wai Chee Shiu, Counting Simsun Permutations by Descents, Ann. Comb. 15, 625-635 (2011). See p. 627.

Ming-Jian Ding and Bao-Xuan Zhu, Some results related to Hurwitz stability of combinatorial polynomials, Advances in Applied Mathematics, Volume 152, (2024), 102591. See p. 35.

R. P. Stanley, Flag f-vectors and the cd-index, Math. Zeitschrift 216 (1994), 483-499.

S. Sundaram, Plethysm, Partitions with an Even Number of Blocks and Euler Numbers, in "Formal Power Series and Algebraic Combinatorics 1994," DIMACS Series in Discrete Mathematics and Theoretical Computer Science 24, AMS (1996).

Formula

T(n, k) = (k+1)*T(n-1, k) + (n-2k+1)*T(n-1, k-1);

Row g.f.: T(n, t) = Sum_{k=0..floor(n/2)} T(n, k)*t^k,

T(n, t) = ((n-1)*t + 1)*T(n-1, t) + t*(1-2t)*T(n-1, t)'.

E.g.f.: Sum_{n>=1} T(n, t)*x^n/n! = (2t-1)*(sec(x*sqrt(2t-1)/2)/(sqrt(2t-1) - tan(x*sqrt(2t-1)/2)))^2.

Example

Triangle begins
   1;
   1,   1;
   1,   4;
   1,  11,   4;
   1,  26,  34;
   1,  57, 180,  34;
   ...

Mathematica

Table[SeriesCoefficient[(2t-1)*(Sec[x*Sqrt[2t-1]/2]/(Sqrt[2t-1]- Tan[x*Sqrt[2t-1]/2]))^2, {x, 0, n}, {t, 0, k}]n!, {n, 11}, {k, 0, Floor[n/2]}]//Flatten (* Stefano Spezia, Aug 09 2023 *)

Crossrefs

Cf. A000111, A000295, A002105.

Sequence in context: A124324 A178519 A094503 * A158753 A183884 A135552

Adjacent sequences: A113894 A113895 A113896 * A113898 A113899 A113900

Keyword

easy,nonn,tabf

Author

Chak-On Chow (cchow(AT)alum.mit.edu), Jan 28 2006

Extensions

Corrected and extended by Vladeta Jovovic, Jan 30 2006

Status

approved