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A047884
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Triangle of numbers a(n,k) = number of Young tableaux with n cells and k rows (1<=k<=n); also number of self-inverse permutations on n letters in which the length of the longest scattered (i.e. not necessarily contiguous) increasing subsequence is k.
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17
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1, 1, 1, 1, 2, 1, 1, 5, 3, 1, 1, 9, 11, 4, 1, 1, 19, 31, 19, 5, 1, 1, 34, 92, 69, 29, 6, 1, 1, 69, 253, 265, 127, 41, 7, 1, 1, 125, 709, 929, 583, 209, 55, 8, 1, 1, 251, 1936, 3356, 2446, 1106, 319, 71, 9, 1, 1, 461, 5336, 11626, 10484, 5323, 1904, 461, 89, 10, 1
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table;
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refs;
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history;
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OFFSET
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1,5
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REFERENCES
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W. Fulton, Young Tableaux, Cambridge, 1997.
D. Stanton and D. White, Constructive Combinatorics, Springer, 1986.
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LINKS
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Alois P. Heinz, Rows n = 1..40, flattened
Index entries for sequences related to Young tableaux.
R. P. Stanley, A combinatorial miscellany
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EXAMPLE
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For n=3 the 4 tableaux are
1 2 3 . 1 2 . 1 3 . 1
. . . . 3 . . 2 . . 2
. . . . . . . . . . 3
Triangle begins:
1;
1, 1;
1, 2, 1;
1, 5, 3, 1;
1, 9, 11, 4, 1;
1, 19, 31, 19, 5, 1;
1, 34, 92, 69, 29, 6, 1;
1, 69, 253, 265, 127, 41, 7, 1;
1, 125, 709, 929, 583, 209, 55, 8, 1;
1, 251, 1936, 3356, 2446, 1106, 319, 71, 9, 1;
1, 461, 5336, 11626, 10484, 5323, 1904, 461, 89, 10, 1;
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MAPLE
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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:
T:= (n, k)-> g(n, k, [])-g(n, k-1, []):
seq (seq (T(n, k), k=1..n), n=1..11); # Alois P. Heinz, Apr 16 2012
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MATHEMATICA
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Table[ Plus@@( NumberOfTableaux/@ Reverse/@Union[ Sort/@(Compositions[ n-m, m ]+1) ]), {n, 12}, {m, n} ]
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CROSSREFS
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Row sums give A000085.
Cf. A049400, A049401, and A178249 which imposes contiguity.
Columns k=1-10 give: A000012, A014495, A217323, A217324, A217325, A217326, A217327, A217328, A217321, A217322. - Alois P. Heinz, Oct 03 2012
Sequence in context: A107735 A137570 A079213 * A124328 A055818 A106240
Adjacent sequences: A047881 A047882 A047883 * A047885 A047886 A047887
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KEYWORD
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nonn,tabl,nice,easy
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AUTHOR
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Wouter Meeussen
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EXTENSIONS
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Definition amended ('scattered' added) by Wouter Meeussen, Dec 22 2010
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STATUS
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approved
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