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A060240
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Triangle T(n,k) in which n-th row gives degrees of irreducible representations of symmetric group S_n.
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19
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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
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OFFSET
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0,7
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COMMENTS
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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)
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REFERENCES
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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.
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LINKS
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EXAMPLE
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Triangle begins:
1;
1;
1, 1;
1, 1, 2;
1, 1, 2, 3, 3;
1, 1, 4, 4, 5, 5, 6; ...
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MAPLE
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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, [])])[]:
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MATHEMATICA
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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 *)
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PROG
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(Magma) CharacterTable(SymmetricGroup(6)); // (say)
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CROSSREFS
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Maximal entry in each row gives A003040.
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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