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
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
Programs
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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