A003876 -id:A003876 - 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

A088961 Zigzag matrices listed entry by entry.

Original entry on oeis.org

3, 5, 5, 5, 10, 14, 14, 7, 14, 21, 21, 7, 21, 35, 42, 48, 27, 9, 48, 69, 57, 36, 27, 57, 78, 84, 9, 36, 84, 126, 132, 165, 110, 44, 11, 165, 242, 209, 121, 55, 110, 209, 253, 220, 165, 44, 121, 220, 297, 330, 11, 55, 165, 330, 462

Offset: 1

Paul Boddington, Oct 28 2003

For each n >= 1 the n X n matrix Z(n) is constructed as follows. The i-th row of Z(n) is obtained by generating a hexagonal array of numbers with 2*n+1 rows, 2*n numbers in the odd numbered rows and 2*n+1 numbers in the even numbered rows. The first row is all 0's except for two 1's in the i-th and the (2*n+1-i)th positions. The remaining rows are generated using the same rule for generating Pascal's triangle. The i-th row of Z(n) then consists of the first n numbers in the bottom row of our array.
For example the top row of Z(2) is [5,5], found from the array:
. 1 0 0 1
1 1 0 1 1
. 2 1 1 2
2 3 2 3 2
. 5 5 5 5
Zigzag matrices have remarkable properties. Here is a selection:
1) Z(n) is symmetric.
2) det(Z(n)) = A085527(n).
3) tr(Z(n)) = A033876(n-1).
4) If 2*n+1 is a power of a prime p then all entries of Z(n) are multiples of p.
5) If 4*n+1 is a power of a prime p then the dot product of any two distinct rows of Z(n) is a multiple of p.
6) It is always possible to move from the bottom left entry of Z(n) to the top right entry using only rightward and upward moves and visiting only odd numbers.
A001700(n) = last term of last row of Z(n): a(A000330(n-1)) = A001700(n); A230585(n) = first term of first row of Z(n): a(A056520(n-1)) = A230585(n); A051417(n) = greatest common divisor of entries of Z(n). - Reinhard Zumkeller, Oct 25 2013

			The first five values are 3, 5, 5, 5, 10 because the first two zigzag matrices are [[3]] and [[5,5],[5,10]].

Reinhard Zumkeller, Matrices Z(n): n = 1..30, flattened

Cf. A085527, A003876.

a088961 n = a088961_list !! (n-1)
a088961_list = concat $ concat $ map f [1..] where
   f x = take x $ g (take x (1 : [0,0..])) where
     g us = (take x $ g' us) : g (0 : init us)
     g' vs = last $ take (2 * x + 1) $
                    map snd $ iterate h (0, vs ++ reverse vs)
   h (p,ws) = (1 - p, drop p $ zipWith (+) ([0] ++ ws) (ws ++ [0]))
-- Reinhard Zumkeller, Oct 25 2013

Flatten[Table[Binomial[2n,n+j-i]-Binomial[2n,n+i+j]+ Binomial[2n, 3n+1-i-j], {n,5},{i,n},{j,n}]] (* Harvey P. Dale, Dec 15 2011 *)

The ij entry of Z(n) is binomial(2*n, n+j-i) - binomial(2*n, n+i+j) + binomial(2*n, 3*n+1-i-j).

A003877 Degrees of irreducible representations of symmetric group S_13.

Original entry on oeis.org

1, 1, 12, 12, 65, 65, 66, 66, 208, 208, 220, 220, 429, 429, 429, 429, 429, 429, 495, 495, 572, 572, 792, 792, 924, 936, 936, 1287, 1287, 1365, 1365, 1430, 1430, 2574, 2574, 2574, 2574, 2860, 2860, 3003, 3003, 3432, 3432, 3432, 3432, 3432, 3432, 3575, 3575, 3640, 3640, 4004, 4004, 4212, 4212, 4290, 4290, 5005, 5005, 5148, 5148, 5720, 5720, 6006, 6006, 6435, 6435, 6864, 6864, 7371, 7371, 7800, 7800, 8580, 8580, 8580, 9009, 9009, 9360, 9360, 10296, 10296, 11440, 11440, 11583, 11583, 12012, 12012, 12012, 12012, 12870, 12870, 15015, 15015, 16016, 17160, 17160, 20592, 20592, 21450, 21450

Offset: 1

N. J. A. Sloane

All 101 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].

Cf. A003869 - A003876, A059796.

Row n=13 of A060240.

A003877 := List(Irr(CharacterTable("S13")), chi->chi[1]);; Sort(A003877); # 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[13]] (* Jean-François Alcover, Sep 23 2024, after Alois P. Heinz in A060240 *)

More terms from Emeric Deutsch, May 13 2004

A093791 Hook products of all partitions of 12.

Original entry on oeis.org

62208, 82944, 82944, 85050, 85050, 107520, 107520, 115200, 115200, 129600, 129600, 134400, 134400, 136080, 136080, 155520, 155520, 161280, 161280, 179200, 179200, 181440, 201600, 201600, 226800, 226800, 228096, 230400, 230400, 248832

Offset: 1

Emeric Deutsch, May 17 2004

Jean-François Alcover, Table of n, a(n) for n = 1..77

Cf. A003876, A060240.

Row n=12 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(12)[q])),q=1..numbpart(12))]);

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[12!/T[12]] (* Jean-François Alcover, Sep 22 2024, after Alois P. Heinz in A060240 *)

a(n) = 12!/A003876(78-n).

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

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A088961 Zigzag matrices listed entry by entry.

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A003877 Degrees of irreducible representations of symmetric group S_13.

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A093791 Hook products of all partitions of 12.

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