A003040 - OEIS

%I M0811 #55 Sep 23 2024 08:25:45

%S 1,1,2,3,6,16,35,90,216,768,2310,7700,21450,69498,292864,1153152,

%T 4873050,16336320,64664600,249420600,1118939184,5462865408,

%U 28542158568,117487079424,547591590000,2474843571200,12760912164000,57424104738000,295284192952320

%N Highest degree of an irreducible representation of symmetric group S_n of degree n.

%C 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

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

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

%H Vasilii Duzhin, <a href="/A003040/b003040.txt">Table of n, a(n) for n = 1..153</a> (terms up to a(80) from Eric M. Schmidt)

%H S. Comét, <a href="http://dx.doi.org/10.1090/S0025-5718-1960-0119451-0">Improved methods to calculate the characters of the symmetric group</a>, Math. Comp. 14 (1960) 104-117.

%H J. McKay, <a href="http://dx.doi.org/10.1090/S0025-5718-1976-0404414-X">The largest degrees of irreducible characters of the symmetric group</a>. Math. Comp. 30 (1976), no. 135, 624-631. (Gives first 75 terms.)

%H J. McKay, <a href="/A003040/a003040a.jpg">Page 1 of 5 pages of tables from Math. Comp. paper</a> [reports 29th term incorrectly]

%H J. McKay, <a href="/A003040/a003040b.jpg">Page 2 of 5 pages of tables from Math. Comp. paper</a>

%H J. McKay, <a href="/A003040/a003040c.jpg">Page 3 of 5 pages of tables from Math. Comp. paper</a>

%H J. McKay, <a href="/A003040/a003040d.jpg">Page 4 of 5 pages of tables from Math. Comp. paper</a>

%H J. McKay, <a href="/A003040/a003040e.jpg">Page 5 of 5 pages of tables from Math. Comp. paper</a>

%H Igor Pak, Greta Panova, and Damir Yeliussizov, <a href="https://arxiv.org/abs/1804.04693">On the largest Kronecker and Littlewood-Richardson coefficients</a>, arXiv:1804.04693 [math.CO], 2018.

%H R. P. Stanley, <a href="/A003277/a003277.pdf">Letter to N. J. A. Sloane, c. 1991</a>

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

%t 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}]];

%t 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}]]];

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

%t Table[Print[n, " ", a[n]]; a[n], {n, 1, 50}] (* _Jean-François Alcover_, Sep 23 2024, after Alois P. Heinz in A060240 *)

%o (Sage)

%o def A003040(n):

%o     res = 1

%o     for P in Partitions(n):

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

%o     return res

%o # _Eric M. Schmidt_, May 07 2013

%Y A117500 gives the corresponding partitions of n.

%Y Cf. A003869, A003870, A003871, A003872, A003873, A003874, A003875, A003876, A003877.

%K nonn,changed

%O 1,3

%A _N. J. A. Sloane_ and _Richard Stanley_

%E Entry revised and extended by _N. J. A. Sloane_, Apr 28 2006

%E a(29) corrected by _Eric M. Schmidt_, May 07 2013