A003040 - OEIS

A003040

Highest degree of an irreducible representation of symmetric group S_n of degree n.
(Formerly M0811)

1, 1, 2, 3, 6, 16, 35, 90, 216, 768, 2310, 7700, 21450, 69498, 292864, 1153152, 4873050, 16336320, 64664600, 249420600, 1118939184, 5462865408, 28542158568, 117487079424, 547591590000, 2474843571200, 12760912164000, 57424104738000, 295284192952320

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,3

COMMENTS

Highest number of standard tableaux of the Ferrers diagrams of the partitions of n. Example: a(4) = 3 because to the partitions 4, 31, 22, 211, and 1111 there correspond 1, 3, 2, 3, and 1 standard tableaux, respectively. - Emeric Deutsch, Oct 02 2015

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 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites].

D. E. Littlewood, The Theory of Group Characters and Matrix Representations of Groups. 2nd ed., Oxford University Press, 1950, p. 265.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

LINKS

Vasilii Duzhin, Table of n, a(n) for n = 1..153 (terms up to a(80) from Eric M. Schmidt)

S. Com�t, Improved methods to calculate the characters of the symmetric group, Math. Comp. 14 (1960) 104-117.

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

J. McKay, Page 1 of 5 pages of tables from Math. Comp. paper [reports 29th term incorrectly]

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

Igor Pak, Greta Panova, and Damir Yeliussizov, On the largest Kronecker and Littlewood-Richardson coefficients, arXiv:1804.04693 [math.CO], 2018.

R. P. Stanley, Letter to N. J. A. Sloane, c. 1991

EXAMPLE

a(5) = 6 because the degrees for S_5 are 1,1,4,4,5,5,6.

MATHEMATICA

h[l_] := 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_] := 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}]]];

a[n_] := a[n] = g[n, n, {}] // Max;

Table[Print[n, " ", a[n]]; a[n], {n, 1, 50}] (* Jean-Fran�ois Alcover, Sep 23 2024, after Alois P. Heinz in A060240 *)

PROG

(Sage)

def A003040(n):

res = 1

for P in Partitions(n):

res = max(res, P.dimension())

return res

# Eric M. Schmidt, May 07 2013

CROSSREFS

A117500 gives the corresponding partitions of n.