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

0,7

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.

Index entries for sequences related to groups

Example

Triangle begins:
  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