A060240 Triangle T(n,k) in which n-th row gives degrees of irreducible representations of symmetric group S_n.

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

Offset: 0

Author

N. J. A. Sloane, Mar 21 2001

Keywords

• nonn
• tabf
• nice
• look
• easy

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)

Examples

			Triangle begins:
  1;
  1;
  1, 1;
  1, 1, 2;
  1, 1, 2, 3, 3;
  1, 1, 4, 4, 5, 5, 6;
  ...
		

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.
• Index entries for sequences related to groups

Crossrefs

Rows give A003870, A003871, etc. Cf. A060241, A060246, A060247.

Maximal entry in each row gives A003040.

Cf. A000041, A000085, A000142, A060437, A093784, A224653.

Programs

Magma

CharacterTable(SymmetricGroup(6)); // (say)

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 *)

Extensions

More terms from Vladeta Jovovic, May 20 2003