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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. 35
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 number of 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

Alois P. Heinz, Antidiagonals n = 0..50, flattened

Wikipedia, Young tableau

FORMULA

Conjecture: A(n,k) ~ k^n/Pi^(k/2) * (k/n)^(k*(k-1)/4) * prod(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-Fran├žois Alcover, Dec 06 2013, translated from Maple *)

CROSSREFS

Columns k=0-12 give: A000007, A000012, A001405, A001006, A005817, A049401, A007579, A007578, A007580, A212915, A212916, A229053, A229068.

Diagonal gives: A000085.

Cf. A047884, A049400, A226873, A240608.

Sequence in context: A109702 A115412 A247506 * A143841 A035440 A029878

Adjacent sequences:  A182169 A182170 A182171 * A182173 A182174 A182175

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Apr 16 2012

STATUS

approved

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Last modified March 30 20:27 EDT 2015. Contains 256050 sequences.