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%I #52 Feb 17 2022 06:17:35
%S 1,1,1,1,1,1,2,1,1,2,3,3,1,1,4,4,5,5,6,1,1,5,5,5,5,9,9,10,10,16,1,1,6,
%T 6,14,14,14,14,15,15,20,21,21,35,35,1,1,7,7,14,14,20,20,21,21,28,28,
%U 35,35,42,56,56,64,64,70,70,90,1,1,8,8,27,27,28,28,42,42,42,48,48,56,56,70,84
%N Triangle T(n,k) in which n-th row gives degrees of irreducible representations of symmetric group S_n.
%C Sum_{k>=1} T(n,k)^2 = n!. - _R. J. Mathar_, May 09 2013
%C From _Emeric Deutsch_, Oct 31 2014: (Start)
%C Number of entries in row n = A000041(n) = number of partitions of n.
%C Sum of entries in row n = A000085(n).
%C Largest (= last) entry in row n = A003040(n).
%C The entries in row n give the number of standard Young tableaux of the Ferrers diagrams of the partitions of n (nondecreasingly). (End)
%D J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups, Oxford Univ. Press, 1985.
%D B. E. Sagan, The Symmetric Group, 2nd ed., Springer, 2001, New York.
%H Alois P. Heinz, <a href="/A060240/b060240.txt">Rows n = 0..36, flattened</a>
%H J. S. Frame, G. de B. Robinson, and R. M. Thrall, <a href="http://dx.doi.org/10.4153/CJM-1954-030-1">The hook graphs of the symmetric group</a> Canad. J. Math, 6:316-324, 1954. See Theorem 1, p. 318.
%H <a href="/index/Gre#groups">Index entries for sequences related to groups</a>
%e Triangle begins:
%e 1;
%e 1;
%e 1, 1;
%e 1, 1, 2;
%e 1, 1, 2, 3, 3;
%e 1, 1, 4, 4, 5, 5, 6; ...
%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, n, [])])[]:
%p seq(T(n), n=0..10); # _Alois P. Heinz_, Jan 07 2013
%t h[l_List] := 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_List] := If[n == 0 || i == 1, h[Join[l, Array[1&, n]]], If[i<1, 0, Flatten @ Table[g[n-i*j, i-1, Join[l, Array[i&, j]]], {j, 0, n/i}]]]; T[n_] := Sort[g[n, n, {}]]; T[1] = {1}; Table[T[n], {n, 1, 10}] // Flatten (* _Jean-François Alcover_, Jan 27 2014, after _Alois P. Heinz_ *)
%o (Magma) CharacterTable(SymmetricGroup(6)); // (say)
%Y Rows give A003870, A003871, etc. Cf. A060241, A060246, A060247.
%Y Maximal entry in each row gives A003040.
%Y Cf. A000041, A000085, A000142, A060437, A224653.
%K nonn,tabf,nice,look,easy
%O 0,7
%A _N. J. A. Sloane_, Mar 21 2001
%E More terms from _Vladeta Jovovic_, May 20 2003
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