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.

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

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

R. P. Stanley, A combinatorial miscellany

Index entries for sequences related to Young tableaux.

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} ]
(* Second program: *)
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-François 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 A368487 A055818

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