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

1,2

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.

Formula

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

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) *)

Crossrefs

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

Column k=7 of A214015.

Sequence in context: A360384 A226439 A248840 * A230051 A067455 A033646

Adjacent sequences: A072128 A072129 A072130 * A072132 A072133 A072134

Keyword

nonn

Author

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

Extensions

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

Status

approved