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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!.

%I A224653 #23 Aug 30 2018 22:08:11
%S A224653 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 A224653 28,35,42,56,64,70,90,1,8,27,28,42,48,56,70,84,105,120,162,168,189,
%U A224653 216,1,9,35,36,42,75,84,90,126,160,210,225,252,288,300,315,350,448,450,525,567,768
%N A224653 Irregular table which shows in row n the dimensions of the irreducible representations of the permutation group of order n!.
%C A224653 This is triangle A060240 if duplicates in individual rows are removed.
%C A224653 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 A224653 Alois P. Heinz, <a href="/A224653/b224653.txt">Rows n = 0..40, flattened</a>
%e A224653 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 A224653 1;
%e A224653 1;
%e A224653 1;
%e A224653 1,2;
%e A224653 1,2,3;
%e A224653 1,4,5,6;
%e A224653 1,5,9,10,16;
%e A224653 1,6,14,15,20,21,35;
%e A224653 1,7,14,20,21,28,35,42,56,64,70,90;
%e A224653 1,8,27,28,42,48,56,70,84,105,120,162,168,189,216;
%e A224653 1,9,35,36,42,75,84,90,126,160,210,225,252,288,300,315,350,448,450,525,567,768;
%p A224653 h:= proc(l) local n; n:= nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
%p A224653       add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n) end:
%p A224653 g:= (n, i, l)-> `if`(n=0 or i=1, h([l[], 1$n]), `if`(i<1, 0,
%p A224653                  seq(g(n-i*j, i-1, [l[], i$j]), j=0..n/i))):
%p A224653 T:= n-> sort([{g(n$2, [])}[]])[]:
%p A224653 seq(T(n), n=0..12);  # _Alois P. Heinz_, May 26 2015
%t A224653 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 A224653 Cf. A060240.
%K A224653 nonn,tabf
%O A224653 0,5
%A A224653 _R. J. Mathar_, May 09 2013