A072132 T_8(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, 40320, 362879, 3628718, 39912738, 478842196, 6221523082, 87002638276, 1302313974900, 20763508263000, 351019617373500, 6266271456118776, 117671982989344680, 2316256222907194304, 47635421509263043024, 1020455890785584587168

Offset

1,2

Links

Alois P. Heinz, Table of n, a(n) for n = 1..586

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.

Formula

a(n) ~ 1913625 * 2^(6*n + 77) / (n^(63/2) * Pi^(7/2)). - Vaclav Kotesovec, Sep 10 2014

Maple

a:= proc(n) option remember; `if`(n<4, n!,
      (-147456*(n+4)*(n-1)^2*(n-2)^2*(n-3)^2*a(n-4)
      +128*(33876+30709*n+6687*n^2+410*n^3)*(n-1)^2*(n-2)^2*a(n-3)
      -4*(1092*n^5+37140*n^4+455667*n^3+2387171*n^2+4649270*n+1206000)*
      (n-1)^2*a(n-2) +(-17075520+(22488312+(29223280+(10509820+(1764252+
      (154164+(6804+120*n)*n)*n)*n)*n)*n)*n)*a(n-1))/
      ((n+16)*(n+7)^2*(n+15)^2*(n+12)^2))
    end:
seq(a(n), n=1..30);  # Alois P. Heinz, Sep 28 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 <= 8, n!, g[n, 8, {}]]; 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=8 of A214015.

Sequence in context: A193936 A226440 A248841 * A230231 A066459 A269221

Adjacent sequences: A072129 A072130 A072131 * A072133 A072134 A072135

Keyword

nonn

Author

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

Status

approved