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A047884 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. 18
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

REFERENCES

W. Fulton, Young Tableaux, Cambridge, 1997.

D. Stanton and D. White, Constructive Combinatorics, Springer, 1986.

LINKS

Alois P. Heinz, Rows n = 1..68, flattened

Index entries for sequences related to Young tableaux.

R. P. Stanley, A combinatorial miscellany

EXAMPLE

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;

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) `if`(n=0 or i=1, (p->h(p)*x^`if`(p=[], 0, p[1]))

      ([l[], 1$n]), add(g(n-i*j, i-1, [l[], i$j]), j=0..n/i))

    end:

T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(g(n$2, [])):

seq(T(n), n=1..14); # Alois P. Heinz, Apr 16 2012, revised Mar 05 2014

MATHEMATICA

Table[ Plus@@( NumberOfTableaux/@ Reverse/@Union[ Sort/@(Compositions[ n-m, m ]+1) ]), {n, 12}, {m, n} ]

h[l_] := With[{n=Length[l]}, Total[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_] := If[n== 0|| i==1, Function[p, h[p]*x^If[p == {}, 0, p[[1]] ] ] [ Join[l, Array[1&, n]]], Sum[g[n-i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, n}]][g[n, n, {}]]; Table[T[n], {n, 1, 14}] // Flatten (* Jean-Fran├žois Alcover, Oct 26 2015, after Alois P. Heinz *)

CROSSREFS

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

a(2n,n) gives A267436.

Sequence in context: A107735 A137570 A079213 * A124328 A055818 A106240

Adjacent sequences:  A047881 A047882 A047883 * A047885 A047886 A047887

KEYWORD

nonn,tabl,nice,easy

AUTHOR

Wouter Meeussen

EXTENSIONS

Definition amended ('scattered' added) by Wouter Meeussen, Dec 22 2010

STATUS

approved

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Last modified May 24 09:45 EDT 2016. Contains 273237 sequences.