A239119 Number of ballot sequences of length n with exactly 8 fixed points.

0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 3, 9, 29, 99, 357, 1351, 5343, 21992, 93912, 414848, 1891264, 8878972, 42849860, 212214460, 1077052284, 5594301872, 29704267536, 161055535088, 890880956848, 5022885935600, 28843306388880, 168562494708400, 1001888980299056

Offset: 0

Author

Joerg Arndt and Alois P. Heinz, Mar 10 2014

Keywords

• nonn

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

Examples

			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].
		

Links

• Joerg Arndt and Alois P. Heinz, Table of n, a(n) for n = 0..800
• Wikipedia, Young tableau

Crossrefs

Column k=8 of A238802.

Programs

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 *)

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