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A182172 Number A(n,k) of standard Young tableaux of n cells and height <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals. 44
1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1, 1, 2, 3, 1, 0, 1, 1, 2, 4, 6, 1, 0, 1, 1, 2, 4, 9, 10, 1, 0, 1, 1, 2, 4, 10, 21, 20, 1, 0, 1, 1, 2, 4, 10, 25, 51, 35, 1, 0, 1, 1, 2, 4, 10, 26, 70, 127, 70, 1, 0, 1, 1, 2, 4, 10, 26, 75, 196, 323, 126, 1, 0, 1, 1, 2, 4, 10, 26, 76, 225, 588, 835, 252, 1, 0 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,13
COMMENTS
Also the number A(n,k) of standard Young tableaux of n cells and <= k columns.
A(n,k) is also the number of n-length words w over a k-ary alphabet {a1,a2,...,ak} such that for every prefix z of w we have #(z,a1) >= #(z,a2) >= ... >= #(z,ak), where #(z,x) counts the letters x in word z. The A(4,4) = 10 words of length 4 over alphabet {a,b,c,d} are: aaaa, aaab, aaba, abaa, aabb, abab, aabc, abac, abca, abcd.
LINKS
Wikipedia, Young tableau
FORMULA
Conjecture: A(n,k) ~ k^n/Pi^(k/2) * (k/n)^(k*(k-1)/4) * Product_{j=1..k} Gamma(j/2). - Vaclav Kotesovec, Sep 12 2013
EXAMPLE
A(4,2) = 6, there are 6 standard Young tableaux of 4 cells and height <= 2:
+------+ +------+ +---------+ +---------+ +---------+ +------------+
| 1 3 | | 1 2 | | 1 3 4 | | 1 2 4 | | 1 2 3 | | 1 2 3 4 |
| 2 4 | | 3 4 | | 2 .-----+ | 3 .-----+ | 4 .-----+ +------------+
+------+ +------+ +---+ +---+ +---+
Square array A(n,k) begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 2, 2, 2, 2, 2, 2, ...
0, 1, 3, 4, 4, 4, 4, 4, 4, ...
0, 1, 6, 9, 10, 10, 10, 10, 10, ...
0, 1, 10, 21, 25, 26, 26, 26, 26, ...
0, 1, 20, 51, 70, 75, 76, 76, 76, ...
0, 1, 35, 127, 196, 225, 231, 232, 232, ...
0, 1, 70, 323, 588, 715, 756, 763, 764, ...
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, 0, `if`(i=1, h([l[], 1$n]),
g(n, i-1, l) +`if`(i>n, 0, g(n-i, i, [l[], i])))))
end:
A:= (n, k)-> g(n, k, []):
seq(seq(A(n, d-n), n=0..d), d=0..15);
MATHEMATICA
h[l_List] := Module[{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_List] := g[n, i, l] = If[n == 0, h[l], If[i < 1, 0, If[i == 1, h[Join[l, Array[1&, n]]], g [n, i-1, l] + If[i > n, 0, g[n-i, i, Append[l, i]]]]]];
a[n_, k_] := g[n, k, {}];
Table[Table[a[n, d-n], {n, 0, d}], {d, 0, 15}] // Flatten (* Jean-François Alcover, Dec 06 2013, translated from Maple *)
CROSSREFS
Main diagonal gives A000085.
A(2n,n) gives A293128.
Sequence in context: A293285 A262553 A247506 * A143841 A276719 A276837
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Apr 16 2012
STATUS
approved

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Last modified March 19 01:22 EDT 2024. Contains 370952 sequences. (Running on oeis4.)