A003872 -id:A003872 - cp's OEIS frontend

A003040 Highest degree of an irreducible representation of symmetric group S_n of degree n.

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

Offset: 1

N. J. A. Sloane and Richard Stanley

nonn

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

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

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).

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

A117500 gives the corresponding partitions of n.

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

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-François Alcover, Sep 23 2024, after Alois P. Heinz in A060240 *)

def A003040(n):
    res = 1
    for P in Partitions(n):
        res = max(res, P.dimension())
    return res
# Eric M. Schmidt, May 07 2013

Entry revised and extended by N. J. A. Sloane, Apr 28 2006

a(29) corrected by Eric M. Schmidt, May 07 2013

A003869 Degrees of irreducible representations of symmetric group S_5.

Original entry on oeis.org

1, 1, 4, 4, 5, 5, 6

Offset: 1

N. J. A. Sloane

All 7 terms of this finite sequence are shown.

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].

Row n=5 of A060240.

Cf. A003870, A003871, A003872, A003873.

A003869 := List(Irr(CharacterTable("S5")), chi->chi[1]);; Sort(A003869); # Eric M. Schmidt, Jul 18 2012

Magma
```
// See A003875 for Magma code
```

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}]]];
T[n_] := g[n, n, {}];
Sort[T[5]] (* Jean-François Alcover, Sep 22 2024, after Alois P. Heinz in A060240 *)

A093786 Hook products of all partitions of 8.

Original entry on oeis.org

448, 576, 576, 630, 630, 720, 720, 960, 1152, 1152, 1440, 1440, 1920, 1920, 2016, 2016, 2880, 2880, 5760, 5760, 40320, 40320

Offset: 1

Emeric Deutsch, May 17 2004

Cf. A003872, A060240.

Row n=8 of A093784.

H:=proc(pa) local F,j,p,Q,i,col,a,A: F:=proc(x) local i, ct: ct:=0: for i from 1 to nops(x) do if x[i]>1 then ct:=ct+1 else fi od: ct; end: for j from 1 to nops(pa) do p[1][j]:=pa[j] od: Q[1]:=[seq(p[1][j],j=1..nops(pa))]: for i from 2 to pa[1] do for j from 1 to F(Q[i-1]) do p[i][j]:=Q[i-1][j]-1 od: Q[i]:=[seq(p[i][j],j=1..F(Q[i-1]))] od: for i from 1 to pa[1] do col[i]:=[seq(Q[i][j]+nops(Q[i])-j,j=1..nops(Q[i]))] od: a:=proc(i,j) if i<=nops(Q[j]) and j<=pa[1] then Q[j][i]+nops(Q[j])-i else 1 fi end: A:=matrix(nops(pa),pa[1],a): product(product(A[m,n],n=1..pa[1]),m=1..nops(pa)); end: with(combinat): rev:=proc(a) [seq(a[nops(a)+1-i],i=1..nops(a))] end: sort([seq(H(rev(partition(8)[q])),q=1..numbpart(8))]);

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}]]];
T[n_] := g[n, n, {}];
Sort[8!/T[8]] (* Jean-François Alcover, Aug 12 2024, after Alois P. Heinz in A060240 *)

a(n) = 8!/A003872(23-n).

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A003040 Highest degree of an irreducible representation of symmetric group S_n of degree n.

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A003869 Degrees of irreducible representations of symmetric group S_5.

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A093786 Hook products of all partitions of 8.

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