Voozh

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

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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, 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} ]

(* Alternative: *)

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.