A224653 Irregular table which shows in row n the dimensions of the irreducible representations of the permutation group of order n!.

1, 1, 1, 1, 2, 1, 2, 3, 1, 4, 5, 6, 1, 5, 9, 10, 16, 1, 6, 14, 15, 20, 21, 35, 1, 7, 14, 20, 21, 28, 35, 42, 56, 64, 70, 90, 1, 8, 27, 28, 42, 48, 56, 70, 84, 105, 120, 162, 168, 189, 216, 1, 9, 35, 36, 42, 75, 84, 90, 126, 160, 210, 225, 252, 288, 300, 315, 350, 448, 450, 525, 567, 768

Offset: 0

Author

R. J. Mathar, May 09 2013

Keywords

• nonn
• tabf

Comments

This is triangle A060240 if duplicates in individual rows are removed.

The entries in row n give the number of standard Young tableaux of the Ferrers diagrams of the partitions of n (without duplicates, increasingly). Example: n = 4; there are 5 partitions of 4: [4], [3,1], [2,2], [2,1,1], and [1,1,1,1]; their Ferrers graphs have 1, 3, 2, 3, and 1 standard tableaux, respectively. - Emeric Deutsch, May 26 2015

Examples

			The group of permutations of [8] has 2 representations of dimension 1, 2 of dimension 7, 2 of dimension 14, 2 of dimension 20, 2 of dimension 21, 2 of dimension 28, 2 of dimension 35, 1 of dimension 42, 2 of dimension 56, 2 of dimension 64, 2 of dimension of 70 and 1 of dimension 90.
1;
1;
1;
1,2;
1,2,3;
1,4,5,6;
1,5,9,10,16;
1,6,14,15,20,21,35;
1,7,14,20,21,28,35,42,56,64,70,90;
1,8,27,28,42,48,56,70,84,105,120,162,168,189,216;
1,9,35,36,42,75,84,90,126,160,210,225,252,288,300,315,350,448,450,525,567,768;
		

Links

• Alois P. Heinz, Rows n = 0..40, flattened

Crossrefs

Cf. A060240.

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:= (n, i, l)-> `if`(n=0 or i=1, h([l[], 1$n]), `if`(i<1, 0,
                 seq(g(n-i*j, i-1, [l[], i$j]), j=0..n/i))):
T:= n-> sort([{g(n$2, [])}[]])[]:
seq(T(n), n=0..12);  # Alois P. Heinz, May 26 2015

Mathematica

h[l_List] := Module[{n}, 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_List] := If[n==0 || i==1, h[Join[l, Array[1&, n]]], If[i<1, 0,Table[g[n - i*j, i-1, Join [l, Array[i&, j]]], {j, 0, n/i}]]]; T[n_] := g[n, n, {}] // Flatten // Union; T[1] = {1}; Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Jul 03 2015, after Alois P. Heinz *)