OFFSET
0,3
COMMENTS
Previous name was: Related to Young tableaux of bounded height.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..250
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.
Nathaniel Shar, Experimental methods in permutation patterns and bijective proof, PhD Dissertation, Mathematics Department, Rutgers University, May 2016.
FORMULA
a(n) ~ 5 * 2^(2*n + 6) * 3^(2*n + 21) / (n^(35/2) * Pi^(5/2)). - Vaclav Kotesovec, Sep 10 2014
MAPLE
h:= proc(l) local n; n:=nops(l); add(i, i=l)! /mul(mul(1+l[i]-j
+add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n)
end:
g:= proc(n, i, l) option remember;
`if`(n=0, h(l)^2, `if`(i<1, 0, `if`(i=1, h([l[], 1$n])^2,
g(n, i-1, l)+ `if`(i>n, 0, g(n-i, i, [l[], i])))))
end:
a:= n-> g(n, 6, []):
seq(a(n), n=0..25); # Alois P. Heinz, Apr 10 2012
# second Maple program
a:= proc(n) option remember; `if`(n<7, n!,
((56*n^5-9408+11032*n+19028*n^2+7360*n^3+1092*n^4)*a(n-1)
-4*(196*n^3+1608*n^2+3167*n+444)*(n-1)^2*a(n-2)
+1152*(2*n+3)*(n-1)^2*(n-2)^2*a(n-3))/ ((n+9)*(n+8)^2*(n+5)^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_, k_] := If[k >= n, n!, g[n, k, {}]]; Table[a[n, 6], {n, 0, 30}] (* Jean-François Alcover, Mar 11 2014, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn
AUTHOR
N. J. A. Sloane, Mar 13 2000
EXTENSIONS
More terms from Alois P. Heinz, Apr 10 2012
New name from Vaclav Kotesovec, Sep 10 2014
STATUS
approved