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A224653 Irregular table which shows in row n the dimensions of the irreducible representations of the permutation group of order n!. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,5

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

LINKS

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

EXAMPLE

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;

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-Fran├žois Alcover, Jul 03 2015, after Alois P. Heinz *)

CROSSREFS

Cf. A060240.

Sequence in context: A055889 A125930 A210790 * A101391 A327632 A117704

Adjacent sequences:  A224650 A224651 A224652 * A224654 A224655 A224656

KEYWORD

nonn,tabf

AUTHOR

R. J. Mathar, May 09 2013

STATUS

approved

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Last modified November 14 20:11 EST 2019. Contains 329129 sequences. (Running on oeis4.)