A117500 Triangle read by rows in which row n gives the partition of n associated with highest degree representation of symmetric group S_n.

1, 2, 2, 1, 3, 1, 3, 1, 1, 3, 2, 1, 4, 2, 1, 4, 2, 1, 1, 4, 3, 1, 1, 4, 3, 2, 1, 5, 3, 2, 1, 5, 3, 2, 1, 1, 5, 4, 2, 1, 1, 6, 4, 2, 1, 1, 5, 4, 3, 2, 1, 6, 4, 3, 2, 1, 6, 4, 3, 2, 1, 1, 7, 4, 3, 2, 1, 1, 7, 5, 3, 2, 1, 1, 7, 5, 3, 2, 2, 1, 7, 5, 3, 2, 2, 1, 1, 7, 5, 4, 3, 2, 1, 7, 5, 4, 3, 2, 1

Offset

1,2

Comments

Note that a partition and its conjugate give the same degree representation of the symmetric group. We take the lexicographically earlier of the two.

Links

Table of n, a(n) for n=1..99.

J. McKay, The largest degrees of irreducible characters of the symmetric group. Math. Comp. 30 (1976), no. 135, 624-631. (Gives first 75 rows on pp. 627-629.)

J. McKay, Page 1 of 5 pages of tables from Math. Comp. paper

J. McKay, Page 2 of 5 pages of tables from Math. Comp. paper

J. McKay, Page 3 of 5 pages of tables from Math. Comp. paper

J. McKay, Page 4 of 5 pages of tables from Math. Comp. paper

J. McKay, Page 5 of 5 pages of tables from Math. Comp. paper

Formula

If p_1 >= p_2 >= ... >= p_k is the partition of n, the degree of the representation (given in A003040) is n! * Product_{i<j} (b_i - b_j) / Product_i (b_i!), where b_i = p_i+k-i.

Example

Triangle begins:
1
2
2 1
3 1
3 1 1
3 2 1
4 2 1
4 2 1 1
4 3 1 1
4 3 2 1
5 3 2 1
5 3 2 1 1
5 4 2 1 1
6 4 2 1 1
5 4 3 2 1
6 4 3 2 1
6 4 3 2 1 1
7 4 3 2 1 1
7 5 3 2 1 1
7 5 3 2 2 1
7 5 3 2 2 1 1
7 5 4 3 2 1
7 5 4 3 2 1 1
8 5 4 3 2 1 1
8 6 4 3 2 1 1

Crossrefs

See A003040 for much more information. Cf. A060240.

Sequence in context: A117470 A070786 A255542 * A297626 A206441 A175245

Adjacent sequences: A117497 A117498 A117499 * A117501 A117502 A117503

Keyword

nonn,tabf

Author

N. J. A. Sloane, Apr 28 2006

Status

approved