A113897 - OEIS

%I #34 Aug 05 2024 14:03:08

%S 1,1,1,1,4,1,11,4,1,26,34,1,57,180,34,1,120,768,496,1,247,2904,4288,

%T 496,1,502,10194,28768,11056,1,1013,34096,166042,141584,11056,1,2036,

%U 110392,868744,1372088,349504,1,4083,349500,4247720,11204160,6213288,349504

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

%C Is this A094503 after removal of the top row? - _R. J. Mathar_, Aug 13 2008

%C Yes. See formula of _Peter Bala_, Jun 26 2012 in A094503. - _Stefano Spezia_, Aug 09 2023

%H Chak-On Chow and Wai Chee Shiu, <a href="https://doi.org/10.1007/s00026-011-0113-6">Counting Simsun Permutations by Descents</a>, Ann. Comb. 15, 625-635 (2011). See p. 627.

%H Ming-Jian Ding and Bao-Xuan Zhu, <a href="https://doi.org/10.1016/j.aam.2023.102591">Some results related to Hurwitz stability of combinatorial polynomials</a>, Advances in Applied Mathematics, Volume 152, (2024), 102591. See p. 35.

%H R. P. Stanley, <a href="https://math.mit.edu/~rstan/pubs/pubfiles/96.pdf">Flag f-vectors and the cd-index</a>, Math. Zeitschrift 216 (1994), 483-499.

%H S. Sundaram, <a href="http://citeseerx.ist.psu.edu/pdf/08428845a2d7735bf0d5467a3b51ef55de495e3b">Plethysm, Partitions with an Even Number of Blocks and Euler Numbers</a>, in "Formal Power Series and Algebraic Combinatorics 1994," DIMACS Series in Discrete Mathematics and Theoretical Computer Science 24, AMS (1996).

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

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

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

%F 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.

%e Triangle begins

%e    1;

%e    1,   1;

%e    1,   4;

%e    1,  11,   4;

%e    1,  26,  34;

%e    1,  57, 180,  34;

%e    ...

%t 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 *)

%Y Cf. A000111, A000295, A002105.

%K easy,nonn,tabf

%O 1,5

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

%E Corrected and extended by _Vladeta Jovovic_, Jan 30 2006