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

Author

Wouter Meeussen

Keywords

• nonn
• tabl
• nice
• easy

Examples

			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;
  ...
		

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.

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.

Programs

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 *)

Extensions

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