A060240 -id:A060240 - cp's OEIS frontend

A003876 Degrees of irreducible representations of symmetric group S_12.

1, 1, 11, 11, 54, 54, 55, 55, 132, 132, 154, 154, 165, 165, 275, 275, 297, 297, 320, 320, 330, 330, 462, 462, 462, 462, 616, 616, 891, 891, 945, 945, 1155, 1155, 1320, 1320, 1408, 1408, 1485, 1485, 1650, 1650, 1728, 1728, 1925, 1925, 1925, 1925, 2079, 2079, 2100, 2112, 2112, 2376, 2376, 2640, 2673, 2673, 2970, 2970, 3080, 3080, 3520, 3520, 3564, 3564, 3696, 3696, 4158, 4158, 4455, 4455, 5632, 5632, 5775, 5775, 7700

Offset: 1

N. J. A. Sloane

All 77 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=12 of A060240.

A003876 := List(Irr(CharacterTable("S12")), chi->chi[1]);; Sort(A003876); # 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[12]] (* Jean-François Alcover, Sep 23 2024, after Alois P. Heinz in A060240 *)

A060241 Triangle T(n,k) in which n-th row gives degrees of irreducible representations of alternating group A_n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 3, 1, 3, 3, 4, 5, 1, 5, 5, 8, 8, 9, 10, 1, 6, 10, 10, 14, 14, 15, 21, 35, 1, 7, 14, 20, 21, 21, 21, 28, 35, 45, 45, 56, 64, 70, 1, 8, 21, 21, 27, 28, 35, 35, 42, 48, 56, 84, 105, 120, 162, 168, 189, 216, 1, 9, 35, 36, 42, 75, 84, 90, 126, 160, 210, 224, 224, 225, 252, 288, 300, 315, 350, 384, 384, 450, 525, 567, 1, 10, 44, 45, 110, 120, 126, 126, 132, 165, 210, 231, 330, 385, 462, 550, 594, 594, 594, 660, 693, 825, 924, 990, 990, 1100, 1155, 1232, 1320, 1540, 2310

Offset: 3

N. J. A. Sloane, Mar 21 2001

			Triangle begins:
  1, 1, 1;
  1, 1, 1, 3;
  1, 3, 3, 4, 5;
  1, 5, 5, 8, 8, 9, 10;
  ...

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups, Oxford Univ. Press, 1985.

Rows give A003862, A003863, etc. Cf. A060240.

CharacterTable(AlternatingGroup(6)); // (say)

Characters for A_8 to A_11 from the online Magma calculator. - R. J. Mathar, Jun 04 2006

A060437 a(n) is the number of different degrees in the sequence of the degrees of the irreducible representations of the symmetric group S_n, i.e., count each degree only once.

Original entry on oeis.org

1, 1, 2, 3, 4, 5, 7, 12, 15, 22, 28, 38, 45, 52, 81, 107, 130, 179, 194, 280, 348, 438, 502, 693, 848, 1037, 1274, 1594, 1847, 2473, 2851, 3652, 4271, 5137, 6140, 7995, 9103, 11046, 12978, 16216, 18348, 23153, 26239, 31880, 37582, 45144, 51469, 63571, 71910

Offset: 1

Avi Peretz (njk(AT)netvision.net.il), Apr 07 2001

nonn

The total number of irreducible representations of S_n is the partition function p(n) (sequence A000041) - this is the total number of the degrees counting multiplicities.

Also a(n) = number of distinct values of A153452(m) when A056239(m) is equal to n. - Naohiro Nomoto, Dec 31 2008

			a(6) = 5 because the degrees for S_6 are 1,1,5,5,5,5,9,9,10,10,16 and counting each degree only once only 5 numbers remain: 1,5,9,10,16.

Cf. A000041, A060240, A215366.

with(numtheory):
g:= proc(n) option remember; `if`(n=1, 1,
      add(g(n/q*`if`(q=2, 1, prevprime(q))), q=factorset(n)))
    end:
b:= proc(n, i) option remember; `if`(n=0 or i<2, [2^n],
      [seq(map(p->p*ithprime(i)^j, b(n-i*j, i-1))[], j=0..n/i)])
    end:
a:= n-> nops(map(g, {b(n, n)[]})):
seq(a(n), n=1..30);  # Alois P. Heinz, Aug 09 2012

g[n_] := g[n] = If[n == 1, 1, Sum[g[n/q*If[q == 2, 1, NextPrime[q, -1]]], {q,     FactorInteger[n][[All, 1]]}]]; b[n_, i_] :=b[n, i] = If[n == 0 || i<2, {2^n}, Flatten @ Table[ Map[Function[{p}, p*Prime[i]^j], b[n-i*j, i-1]], {j, 0, n/i}] ]; a[n_] := Length[Union[g /@ b[n, n]]]; Table[a[n], {n, 1, 30}] (* Jean-François Alcover, Apr 15 2015, after Alois P. Heinz *)

More terms from Vladeta Jovovic, May 20 2003

a(22)-a(49) from Alois P. Heinz, Aug 09 2012

A003872 Degrees of irreducible representations of symmetric group S_8.

Original entry on oeis.org

1, 1, 7, 7, 14, 14, 20, 20, 21, 21, 28, 28, 35, 35, 42, 56, 56, 64, 64, 70, 70, 90

Offset: 1

N. J. A. Sloane

All 22 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=8 of A060240.

A003872 := List(Irr(CharacterTable("S8")), chi->chi[1]);; Sort(A003872); # Eric M. Schmidt, Jul 18 2012

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

A003873 Degrees of irreducible representations of symmetric group S_9.

Original entry on oeis.org

1, 1, 8, 8, 27, 27, 28, 28, 42, 42, 42, 48, 48, 56, 56, 70, 84, 84, 105, 105, 120, 120, 162, 162, 168, 168, 189, 189, 216, 216

Offset: 1

N. J. A. Sloane

All 30 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=9 of A060240.

A003873 := List(Irr(CharacterTable("S9")), chi->chi[1]);; Sort(A003873); # Eric M. Schmidt, Jul 18 2012

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

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

A060247 Triangle whose rows are the degrees of the irreducible representations of the groups PSL(2,q) as q runs through the primes and prime powers.

Original entry on oeis.org

1, 1, 2, 1, 1, 1, 3, 1, 3, 3, 4, 5, 1, 3, 3, 4, 5, 1, 3, 3, 6, 7, 8, 1, 7, 7, 7, 7, 8, 9, 9, 9, 1, 5, 5, 8, 8, 9, 10, 1, 5, 5, 10, 10, 11, 12, 12, 1, 7, 7, 12, 12, 12, 13, 14, 14, 1, 15, 15, 15, 15, 15, 15, 15, 15, 16, 17, 17, 17, 17, 17, 17, 17, 1, 9, 9, 16, 16, 16, 16, 17, 18, 18, 18

Offset: 1

N. J. A. Sloane, Mar 22 2001

			Triangle begins:
  1, 1, 2;
  1, 1, 1, 3;
  1, 3, 3, 4, 5;
  1, 3, 3, 4, 5;
  ...
(for q = 2,3,4,5, ...).

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups, Oxford Univ. Press, 1985.

J. S. Kimberley, First 60 rows of A060247 triangle, flattened.

q = A000961(n+1).
Row length sequence is A177744.
Consecutive row sequences from 3rd to 18th are: A003860, A003860, A003879, A003880, A003861, A003882, A003883, A003884, A003885, A003886, A003887, A003888, A003889, A003890, A003891, A003892.
Cf. A060246, A060240, A060241.

Magma
```
CharacterTable(PSL(2,7)); // (say)
```

&cat[[Degree(irred): irred in CharacterTable(PSL(2,q))]: q in [2..17]| IsPrimePower(q)]; // Jason Kimberley, May 22 2010

Extended by Jason Kimberley, May 22 2010

A097522 Triangle read by rows giving the 246 multinomials described by A005651(5) related to Young tableau and Kostka numbers.

Original entry on oeis.org

1, 16, 1, 25, 12, 1, 36, 15, 8, 1, 25, 18, 10, 8, 1, 16, 10, 6, 5, 4, 1, 1, 4, 5, 6, 5, 4, 1

Offset: 1

Alford Arnold, Aug 27 2004

The antidiagonal 1 4 5 6 5 4 1 is also listed in A007837 in ascending order by value: 1 1 4 4 5 5 6. The 246 cases are distributed in A036038 as 1 5 10 20 30 60 120.

			Triangle is
   1;
  16,  1;
  25, 12,  1;
  36, 15,  8,  1;
  25, 18, 10,  8,  1;
  16, 10,  6,  5,  4,  1;
   1,  4,  5,  6,  5,  4,  1;

R. Stanton, Constructive Combinatorics, 19856, page 83.

Cf. A000085, A000142, A003869, A005651, A036038, A060240.

A003874 Degrees of irreducible representations of symmetric group S_10.

Original entry on oeis.org

1, 1, 9, 9, 35, 35, 36, 36, 42, 42, 75, 75, 84, 84, 90, 90, 126, 126, 160, 160, 210, 210, 225, 225, 252, 252, 288, 288, 300, 300, 315, 315, 350, 350, 448, 450, 450, 525, 525, 567, 567, 768

Offset: 1

N. J. A. Sloane

All 42 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=10 of A060240.

A003874 := List(Irr(CharacterTable("S10")), chi->chi[1]);; Sort(A003874); # 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[10]] (* Jean-François Alcover, Sep 23 2024, after Alois P. Heinz in A060240 *)

A059796 Degrees of irreducible representations of symmetric group S_14.

Original entry on oeis.org

1, 1, 13, 13, 77, 77, 78, 78, 273, 273, 286, 286, 429, 429, 560, 560, 637, 637, 715, 715, 1001, 1001, 1001, 1001, 1287, 1287, 1365, 1365, 1716, 1716, 2002, 2002, 2079, 2079, 4368, 4368, 4576, 4576, 4928, 4928, 5733, 5733, 6006, 6006, 6006, 6006, 6006, 6006, 6435

Offset: 1

N. J. A. Sloane, Feb 22 2001

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].

Eric M. Schmidt, Table of n, a(n) for n = 1..135 (complete sequence)

Cf. A003869 - A003877.

Row n=14 of A060240.

List(Irr(CharacterTable("S14")), chi->chi[1]); # Eric M. Schmidt, Jul 18 2012

Magma
```
// See A003875 for MAGMA code.
```

h:= proc(l) local n; n:= nops(l); add(i, i=l)!/mul(mul(1+l[i]-j+
      add(`if`(l[k]>=j, 1, 0), k=i+1..n), j=1..l[i]), i=1..n) end:
g:= (n, i, l)-> `if`(n=0 or i=1, h([l[], 1$n]), `if`(i<1, 0,
                 seq(g(n-i*j, i-1, [l[], i$j]), j=0..n/i))):
sort([g(14$2, [])])[];  # Alois P. Heinz, Sep 23 2024

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[14]] (* Jean-François Alcover, Sep 23 2024, after Alois P. Heinz *)

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A003876 Degrees of irreducible representations of symmetric group S_12.

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A060241 Triangle T(n,k) in which n-th row gives degrees of irreducible representations of alternating group A_n.

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A060437 a(n) is the number of different degrees in the sequence of the degrees of the irreducible representations of the symmetric group S_n, i.e., count each degree only once.

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A003872 Degrees of irreducible representations of symmetric group S_8.

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A003873 Degrees of irreducible representations of symmetric group S_9.

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A003877 Degrees of irreducible representations of symmetric group S_13.

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Extensions

A060247 Triangle whose rows are the degrees of the irreducible representations of the groups PSL(2,q) as q runs through the primes and prime powers.

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Magma

Magma

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A097522 Triangle read by rows giving the 246 multinomials described by A005651(5) related to Young tableau and Kostka numbers.

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