A117500 - OEIS

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

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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:

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