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 A060240 Triangle T(n,k) in which n-th row gives degrees of irreducible representations of symmetric group S_n. 17
 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, 6, 14, 14, 14, 14, 15, 15, 20, 21, 21, 35, 35, 1, 1, 7, 7, 14, 14, 20, 20, 21, 21, 28, 28, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,7 COMMENTS Sum_{k>=1} T(n,k)^2 = n!. - R. J. Mathar, May 09 2013 From Emeric Deutsch, Oct 31 2014: (Start) Number of entries in row n = A000041(n) = number of partitions of n. Sum of entries in row n = A000085(n). Largest (= last) entry in row n = A003040(n). The entries in row n give the number of standard Young tableaux of the Ferrers diagrams of the partitions of n (nondecreasingly). (End) REFERENCES J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups, Oxford Univ. Press, 1985. B. E. Sagan, The Symmetric Group, 2nd ed., Springer, 2001, New  York. LINKS Alois P. Heinz, Rows n = 0..36, flattened J. S. Frame, G. de B. Robinson, and R. M. Thrall, The hook graphs of the symmetric group Canad. J. Math, 6:316-324, 1954. See Theorem 1, p. 318. EXAMPLE 1; 1; 1,1; 1,1,2; 1,1,2,3,3; 1,1,4,4,5,5,6; ... 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, n, [])])[]: seq(T(n), n=0..10);  # Alois P. Heinz, Jan 07 2013 MATHEMATICA 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 *) PROG (MAGMA) CharacterTable(SymmetricGroup(6)); (say) CROSSREFS Rows give A003870, A003871, etc. Cf. A060241, A060246, A060247. Maximal entry in each row gives A003040. Cf. A000041, A000085, A000142, A060437, A224653. Sequence in context: A124287 A253240 A290472 * A153734 A285554 A128495 Adjacent sequences:  A060237 A060238 A060239 * A060241 A060242 A060243 KEYWORD nonn,tabf,nice,look,easy AUTHOR N. J. A. Sloane, Mar 21 2001 EXTENSIONS More terms from Vladeta Jovovic, May 20 2003 STATUS approved

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Last modified October 18 14:04 EDT 2019. Contains 328161 sequences. (Running on oeis4.)