A072131 T_7(n) in the notation of Bergeron et al., u_k(n) in the notation of Gessel: Related to Young tableaux of bounded height.

1, 2, 6, 24, 120, 720, 5040, 40319, 362815, 3626197, 39832877, 476591309, 6162155981, 85494566892, 1264755621000, 19835792076675, 328115505900675, 5698062006852574, 103455252673577866, 1956590161853191160, 38418713005615268760, 780931481835878011620

Offset: 1

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 = 1..200
• F. Bergeron and F. Gascon, Counting Young tableaux of bounded height, J. Integer Sequences, Vol. 3 (2000), #00.1.7.
• Alin Bostan, Andrew Elvey Price, Anthony John Guttmann, Jean-Marie Maillard, Stieltjes moment sequences for pattern-avoiding permutations, arXiv:2001.00393 [math.CO], 2020.
• 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=7 of A214015.

Programs

Maple

a:= proc(n) option remember; `if`(n<8, n!, ((-343035+429858*n
       +238440*n^3+38958*n^4+634756*n^2+2940*n^5+84*n^6)*a(n-1)
       -(1974*n^4+36336*n^3+213240*n^2+407840*n+82425)*(n-1)^2*a(n-2)
       +2*(49875+42646*n+6458*n^2)*(n-1)^2*(n-2)^2*a(n-3)
       -11025*(n-1)^2*(n-2)^2*(n-3)^2*a(n-4))/ ((n+6)^2*(n+10)^2*(n+12)^2))
    end:
seq (a(n), n=1..30);  # Alois P. Heinz, Sep 26 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 <= 7, n!, g[n, 7, {}]]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Feb 24 2016, after Alois P. Heinz (A214015) *)

Formula

a(n) ~ 6075 * 7^(2*n + 49/2) / (32768 * n^24 * Pi^3). - Vaclav Kotesovec, Sep 10 2014

Extensions

Typo in title corrected by Joel B. Lewis, Jul 16 2009