OFFSET
0,11
COMMENTS
The fixed points are in the first 8 positions.
Also the number of standard Young tableaux with n cells such that the first column contains 1, 2, ..., 8, but not 9. An alternate definition uses the first row.
Conjecture: Generally, for fixed k is column k of A238802 asymptotic to sqrt(2)/(2*(k+1)*(k-1)!) * exp(sqrt(n)-n/2-1/4) * n^(n/2) * (1 + 7/(24*sqrt(n))), (holds for all k<=10). - Vaclav Kotesovec, Mar 11 2014
LINKS
Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..800
Wikipedia, Young tableau
FORMULA
See Maple program.
Recurrence (for n>=10): (n-9)*(n^8 - 45*n^7 + 1302*n^6 - 34146*n^5 + 562989*n^4 - 4387005*n^3 + 7242668*n^2 + 80535276*n + 148594320)*a(n) = (n^9 - 54*n^8 + 1392*n^7 - 33705*n^6 + 734286*n^5 - 9696141*n^4 + 60317333*n^3 - 48716460*n^2 - 234532332*n - 4007057040)*a(n-1) + (n-10)*(n-8)*(n^8 - 37*n^7 + 1015*n^6 - 27223*n^5 + 410284*n^4 - 2451988*n^3 - 2863260*n^2 + 83948328*n + 232515360)*a(n-2). - Vaclav Kotesovec, Mar 11 2014
a(n) ~ sqrt(2)/90720 * exp(sqrt(n)-n/2-1/4) * n^(n/2) * (1+7/(24*sqrt(n))). - Vaclav Kotesovec, Mar 11 2014
EXAMPLE
a(8) = 1: [1,2,3,4,5,6,7,8].
a(9) = 1: [1,2,3,4,5,6,7,8,1].
a(10) = 3: [1,2,3,4,5,6,7,8,1,1], [1,2,3,4,5,6,7,8,1,2], [1,2,3,4,5,6,7,8,1,9].
a(11) = 9: [1,2,3,4,5,6,7,8,1,1,1], [1,2,3,4,5,6,7,8,1,1,2], [1,2,3,4,5,6,7,8,1,1,9], [1,2,3,4,5,6,7,8,1,2,1], [1,2,3,4,5,6,7,8,1,2,3], [1,2,3,4,5,6,7,8,1,2,9], [1,2,3,4,5,6,7,8,1,9,1], [1,2,3,4,5,6,7,8,1,9,2], [1,2,3,4,5,6,7,8,1,9,10].
MAPLE
b:= proc(n) option remember; `if`(n<3, [1, 1, 3][n+1],
((78*n^4 -18395*n^3 -71700*n^2 +536111*n -6824556)*b(n-1)
+(203*n^5 +3335*n^4 +113400*n^3 +811949*n^2 -2733405*n
+5461380)*b(n-2) +(n-3)*(125*n^4 +21309*n^3 +273479*n^2
+556667*n +1829700)*b(n-3)) /
(203*n^4+1789*n^3+80693*n^2+377071*n-3156156))
end:
a:=n-> `if`(n<8, 0, b(n-8)):
seq(a(n), n=0..40);
MATHEMATICA
b[n_, l_List] := b[n, l] = If[n <= 0, 1, b[n - 1, Append[l, 1]] + Sum[If[i == 1 || l[[i - 1]] > l[[i]], b[n - 1, ReplacePart[l, i -> l[[i]] + 1]], 0], {i, 1, Length[l]}]]; a[n_] := If[n == 8, 1, b[n - 9, {2, 1, 1, 1, 1, 1, 1, 1}]]; a[n_ /; n < 8] = 0; Table[ Print["a(", n, ") = ", an = a[n]]; an, {n, 0, 40}] (* Jean-François Alcover, Feb 06 2015, after Maple *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Joerg Arndt and Alois P. Heinz, Mar 10 2014
STATUS
approved