A072133 T_9(n) in the notation of Bergeron et al., u_k(n) in the notation of Gessel: Related to Young tableaux of bounded height.

1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628799, 39916699, 478995537, 6226736369, 87166698628, 1307240982000, 20907446718225, 355162464899601, 6384776070987990, 121061600999380138, 2413632612087046844, 50453964720806671644, 1102844526263334763556

Offset: 0

Author

Jesse Carlsson (j.carlsson(AT)physics.unimelb.edu.au), Jun 25 2002

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..561
• F. Bergeron and F. Gascon, Counting Young tableaux of bounded height, J. Integer Sequences, Vol. 3 (2000), #00.1.7.
• Shalosh B. Ekhad, Nathaniel Shar, and Doron Zeilberger, The number of 1...d-avoiding permutations of length d+r for SYMBOLIC d but numeric r, arXiv:1504.02513 [math.CO], 2015.
• Ira M. Gessel, Symmetric functions and P-recursiveness, J. Combin. Theory Ser. A 53 (1990), no. 2, 257-285.
• Nathaniel Shar, Experimental methods in permutation patterns and bijective proof, PhD Dissertation, Mathematics Department, Rutgers University, May 2016.

Crossrefs

Cf. A052399 for T_6(n), A047890 for T_5(n), A047889 for T_4(n).

Column k=9 of A214015.

Programs

Maple

a:= proc(n) option remember;
      `if`(n<5, n!, ((-1110790863+(1520978576+(1772290401+(607308786+
       (101671498+(9464664+(500874+(14124+165*n)*n)*n)*n)*n)*n)*n)*n)*a(n-1)
       -(1129886062*n+559908333*n^2+111239576*n^3+10655238*n^4+8778*n^6
       +491700*n^5 [phone])*(n-1)^2*a(n-2) +(258011271+234066216*n
       [phone]*n^2+5463876*n^3 +172810*n^4)*(n-1)^2*(n-2)^2*a(n-3)
       -9*(4070430+1504292*n+117469*n^2)* (n-1)^2*(n-2)^2*(n-3)^2*a(n-4)
       +893025*(n-1)^2*(n-2)^2*(n-3)^2*(n-4)^2*a(n-5)) /
       ((n+20)^2*(n+8)^2*(n+18)^2*(n+14)^2))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 10 2012

Mathematica

h[l_] := With[{n = Length[l]}, Sum[i, {i, l}]!/Product[Product[1 + l[[i]] - j + Sum[If[l[[k]] >= j, 1, 0], {k, i+1, n}], {j, 1, l[[i]]}], {i, 1, n}]]; g[n_, i_, l_] := If[n==0 || i==1, h[Join[l, Array[1 &, n]]]^2, If[i < 1, 0, Sum[g[n - i*j, i - 1, Join[l, Array[i &, j]]], {j, 0, n/i}]]]; a[n_] := If[n <= 9, n!, g[n, 9, {}]]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Feb 24 2016, after Alois P. Heinz (A214015) *)

Formula

a(n) ~ 30625 * 3^(4*n + 90) / (2097152 * n^40 * Pi^4). - Vaclav Kotesovec, Sep 10 2014

Extensions

a(0)=1 prepended by Alois P. Heinz, Feb 09 2017