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

Author

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

Keywords

• easy
• nonn
• tabf

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

Examples

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

Links

• 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).

Crossrefs

Cf. A000111, A000295, A002105.

Programs

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 *)

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.

Extensions

Corrected and extended by Vladeta Jovovic, Jan 30 2006