A212916 Number of standard Young tableaux of n cells and height <= 10.

1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, 35695, 140140, 568360, 2389192, 10338315, 46118592, 211120144, 992316928, 4773362476, 23500234512, 118125854560, 606106812640, 3168660576795, 16872323635132, 91369920670420, 503022250919640, 2811920834508705

Offset: 0

Author

Alois P. Heinz, May 30 2012

Keywords

• nonn

Comments

Number of standard Young tableaux of n cells and <= 10 columns.

Also the number of n-length words w over 10-ary alphabet {a1,a2,...,a10} such that for every prefix z of w we have #(z,a1) >= #(z,a2) >= ... >= #(z,a10), where #(z,x) counts the letters x in word z.

Conjecture: generally (for tableaux with height <= k), a(n) ~ k^n/Pi^(k/2) * (k/n)^(k*(k-1)/4) * prod(j=1..k,Gamma(j/2)); set k=10 for this sequence. - Vaclav Kotesovec, Sep 12 2013

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Column k=10 of A182172.

Programs

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), `if`(i=1, h([l[], 1$n]), `if`(i<1, 0,
        g(n, i-1, l) +`if`(i>n, 0, g(n-i, i, [l[], i])))))
    end:
a:= n-> g(n, 10, []):
seq(a(n), n=0..30);
# second Maple program:
a:= proc(n) option remember;
      `if`(n<6, [1, 1, 2, 4, 10, 26][n+1],
      ((70*n^4+4144*n^3+84986*n^2+685800*n+1656000)*a(n-1)
       +4*(n-1)*(35*n^4+1778*n^3+30106*n^2+184221*n+244350)*a(n-2)
       -8*(n-1)*(n-2)*(518*n^2+11916*n+59265)*a(n-3)
       -16*(n-1)*(n-2)*(n-3)*(259*n^2+4819*n+17355)*a(n-4)
       +21600*(n-1)*(n-2)*(n-3)*(n-4)*a(n-5)
       +14400*(n-5)*(n-1)*(n-2)*(n-3)*(n-4)*a(n-6)) /
       ((n+21)*(n+9)*(n+16)*(n+25)*(n+24)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Oct 12 2012

Mathematica

Flatten[{1,RecurrenceTable[{-14400 (-5+n) (-4+n) (-3+n) (-2+n) (-1+n) a[-6+n]-21600 (-4+n) (-3+n) (-2+n) (-1+n) a[-5+n]+16 (-3+n) (-2+n) (-1+n) (17355+4819 n+259 n^2) a[-4+n]+8 (-2+n) (-1+n) (59265+11916 n+518 n^2) a[-3+n]-4 (-1+n) (244350+184221 n+30106 n^2+1778 n^3+35 n^4) a[-2+n]-2 (828000+342900 n+42493 n^2+2072 n^3+35 n^4) a[-1+n]+(9+n) (16+n) (21+n) (24+n) (25+n) a[n]==0, a[1]==1, a[2]==2, a[3]==4, a[4]==10, a[5]==26, a[6]==76}, a, {n, 20}]}] (* Vaclav Kotesovec, Sep 11 2013 *)

Formula

a(n) ~ 42525/32 * 10^(n+45/2)/(Pi^(5/2)*n^(45/2)). - Vaclav Kotesovec, Sep 11 2013