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A007580 Number of Young tableaux of height <= 8.
(Formerly M1220)
10
1, 1, 2, 4, 10, 26, 76, 232, 764, 2619, 9486, 35596, 139392, 562848, 2352064, 10092160, 44546320, 201158620, 930213752, 4387327088, 21115314916, 103386386516, 515097746072, 2605341147472, 13378787264584, 69622529312665, 367161088308490, 1959294979429380 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Also the number of n-length words w over 8-ary alphabet {a1,a2,...,a8} such that for every prefix z of w we have #(z,a1) >= #(z,a2) >= ... >= #(z,a8), where #(z,x) counts the letters x in word z. - Alois P. Heinz, May 30 2012
REFERENCES
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
F. Bergeron, L. Favreau and D. Krob, Conjectures on the enumeration of tableaux of bounded height, Preprint. (Annotated scanned copy)
F. Bergeron, L. Favreau and D. Krob, Conjectures on the enumeration of tableaux of bounded height, Discrete Math, vol. 139, no. 1-3 (1995), 463-468.
FORMULA
a(n) ~ 135/16 * 8^(n+14)/(Pi^2*n^14). - Vaclav Kotesovec, Sep 11 2013
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, 8, []):
seq(a(n), n=0..30); # Alois P. Heinz, Apr 10 2012
# second Maple program:
a:= proc(n) option remember;
`if`(n<4, [1, 1, 2, 4][n+1],
((40*n^3+1084*n^2+8684*n+18480)*a(n-1)
+16*(n-1)*(5*n^3+107*n^2+610*n+600)*a(n-2)
-1024*(n-1)*(n-2)*(n+6)*a(n-3)
-1024*(n-1)*(n-2)*(n-3)*(n+4)*a(n-4)) /
((n+7)*(n+12)*(n+15)*(n+16)))
end:
seq(a(n), n=0..30); # Alois P. Heinz, Oct 12 2012
MATHEMATICA
RecurrenceTable[{1024 (-3+n) (-2+n) (-1+n) (4+n) a[-4+n]+1024 (-2+n) (-1+n) (6+n) a[-3+n]-16 (-1+n) (600+610 n+107 n^2+5 n^3) a[-2+n]-4 (4620+2171 n+271 n^2+10 n^3) a[-1+n]+(7+n) (12+n) (15+n) (16+n) a[n]==0, a[1]==1, a[2]==2, a[3]==4, a[4]==10}, a, {n, 20}] (* Vaclav Kotesovec, Sep 11 2013 *)
CROSSREFS
Column k=8 of A182172. - Alois P. Heinz, May 30 2012
Sequence in context: A220871 A007578 A239079 * A239080 A212915 A239081
KEYWORD
nonn
AUTHOR
EXTENSIONS
More terms from Alois P. Heinz, Apr 10 2012
STATUS
approved

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Last modified March 18 22:56 EDT 2024. Contains 370952 sequences. (Running on oeis4.)