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Irregular table which shows in row n the dimensions of the irreducible representations of the permutation group of order n!.
2

%I #23 Aug 30 2018 22:08:11

%S 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,

%T 28,35,42,56,64,70,90,1,8,27,28,42,48,56,70,84,105,120,162,168,189,

%U 216,1,9,35,36,42,75,84,90,126,160,210,225,252,288,300,315,350,448,450,525,567,768

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

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

%C 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

%H Alois P. Heinz, <a href="/A224653/b224653.txt">Rows n = 0..40, flattened</a>

%e 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.

%e 1;

%e 1;

%e 1;

%e 1,2;

%e 1,2,3;

%e 1,4,5,6;

%e 1,5,9,10,16;

%e 1,6,14,15,20,21,35;

%e 1,7,14,20,21,28,35,42,56,64,70,90;

%e 1,8,27,28,42,48,56,70,84,105,120,162,168,189,216;

%e 1,9,35,36,42,75,84,90,126,160,210,225,252,288,300,315,350,448,450,525,567,768;

%p h:= proc(l) local n; n:= nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+

%p add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n) end:

%p g:= (n, i, l)-> `if`(n=0 or i=1, h([l[], 1$n]), `if`(i<1, 0,

%p seq(g(n-i*j, i-1, [l[], i$j]), j=0..n/i))):

%p T:= n-> sort([{g(n$2, [])}[]])[]:

%p seq(T(n), n=0..12); # _Alois P. Heinz_, May 26 2015

%t 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_ *)

%Y Cf. A060240.

%K nonn,tabf

%O 0,5

%A _R. J. Mathar_, May 09 2013