A060240 -id:A060240 - cp's OEIS frontend

A060246 Triangle whose rows are the degrees of the irreducible representations of the groups PSL(2,p) as p runs through the primes.

1, 1, 2, 1, 1, 1, 3, 1, 3, 3, 4, 5, 1, 3, 3, 6, 7, 8, 1, 5, 5, 10, 10, 11, 12, 12, 1, 7, 7, 12, 12, 12, 13, 14, 14, 1, 9, 9, 16, 16, 16, 16, 17, 18, 18, 18, 1, 9, 9, 18, 18, 18, 18, 19, 20, 20, 20, 20, 1, 11, 11, 22, 22, 22, 22, 22, 23, 24, 24, 24, 24, 24, 1, 15, 15, 28, 28, 28, 28, 28

Offset: 1

N. J. A. Sloane, Mar 22 2001

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

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 123 rows of A060246 triangle, flattened.

Row length sequence is A124678.
Consecutive row sequences from 3rd to 11th are: A003860, A003879, A003882, A003883, A003885, A003886, A003887, A003890, A003891.
Cf. A060247, A060240, A060241.

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

&cat[[Degree(irred): irred in CharacterTable(PSL(2, p))]: p in PrimesUpTo(30)];

Extended by Jason Kimberley, May 23 2010

A089248 a(n) is the sum of the odd degrees of the irreducible representations of the symmetric group S_n.

Original entry on oeis.org

1, 2, 2, 8, 12, 40, 144, 128, 644, 3504, 7000, 48224, 130992, 861792, 3257600, 32768, 425988, 5833312, 27621672, 415526656, 1987852432, 17674429440, 157807273408, 265515959680, 2848581615344, 30980959604096, 114059874705248, 1365388896050048, 6215927122198944

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Dec 11 2003

nonn

a(n) is divisible by 4 for n >= 4. - Eric M. Schmidt, Apr 28 2013

John McKay, Irreducible representations of odd degree, Journal of Algebra 20, 1972 pages 416-418.

Eric M. Schmidt, Table of n, a(n) for n = 1..200
Eric M. Schmidt, Sage code to compute this sequence

Cf. A000085, A059867.

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] = If[n == 1, 1, Select[g[n, n, {}], OddQ] // Total];
Table[Print[n, " ", a[n]];
a[n], {n, 1, 50}] (* Jean-François Alcover, Sep 23 2024, after Alois P. Heinz in A060240 *)

# Simple but inefficient; see links for faster code
def A089248(n) :
    res = 0
    for P in Partitions(n) :
        deg = P.dimension()
        if is_odd(deg) : res += deg
    return res
# Eric M. Schmidt, Apr 28 2013

a(2^n) = 2^(2^n - 1). - Eric M. Schmidt, Apr 28 2013

More terms from Eric M. Schmidt, Apr 28 2013

A104707 Triangle read by rows distributing the 1602 multinomials described by A005651(6) related to Young tableau and Kostka numbers.

Original entry on oeis.org

1, 25, 1, 81, 20, 1, 25, 54, 15, 1, 100, 15, 36, 15, 1, 256, 60, 10, 27, 10, 1, 25, 128, 30, 5, 27, 10, 1, 100, 10, 64, 30, 5, 18, 10, 1, 81, 40, 5, 32, 10, 5, 9, 5, 1, 25, 27, 10, 0, 32, 10, 0, 9, 5, 1, 1, 5, 9, 10, 5, 16, 10, 5, 9, 5, 1

Offset: 1

Alford Arnold, Mar 19 2005

The last row (and the square roots of the first column) is 1 5 9 5 10 16 5 10 9 5 1, which when sorted appears as 1 1 5 5 5 5 9 9 10 10 16 in A003870 and A060240. The 1602 cases are distributed in A036038: 1 6 15 20 30 60 90 120 180 360 720.

			The triangle is:
1;
25,   1;
81,  20,  1;
25,  54, 15,  1;
100, 15, 36, 15,  1;
256, 60, 10, 27, 10,  1;
25, 128, 30,  5, 27, 10,  1;
100, 10, 64, 30,  5, 18, 10, 1;
81,  40,  5, 32, 10,  5,  9, 5, 1;
25,  27, 10,  0, 32, 10,  0, 9, 5, 1;
1,    5,  9, 10,  5, 16, 10, 5, 9, 5, 1;

D. Stanton and D. White, Constructive Combinatorics, 1986, page 83.

Cf. A000041, A000085, A000142, A003870, A060240, A005651, A036038 and A097522 (a similar triangle distributing the 246 multinomials in A005651(5)).

Cf. A104778.

A060369 a(n) is the maximum number of occurrences of a degree in the sequence of the degrees of the irreducible representations of the symmetric group S_n.

Original entry on oeis.org

1, 2, 2, 2, 2, 4, 4, 2, 3, 2, 4, 4, 6, 6, 6, 4, 8, 6, 10, 6, 8, 8, 12, 8, 12, 12, 10, 12, 22, 14, 12, 12, 16, 18, 30, 14, 20, 26, 16, 20, 22, 20, 26, 25, 24, 24, 32, 16, 32, 30, 26, 24, 32, 32, 40, 32, 34, 32, 32, 34, 44, 30, 44, 36, 52, 34, 54, 38, 56, 50

Offset: 1

Avi Peretz (njk(AT)netvision.net.il), Apr 01 2001

nonn

			a(6) = 4 because the degrees for S_6 are 1,1,5,5,5,5,9,9,10,10,16 and the number 5 appears 4 times.

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] = If[n == 1, 1, MaximalBy[Tally[g[n, n, {}]], Last][[1, 2]]];
Table[Print[n, " ", a[n]];
a[n], {n, 1, 50}] (* Jean-François Alcover, Sep 23 2024, after Alois P. Heinz in A060240 *)

def A060369(n) :
    mult = {}
    for P in Partitions(n):
        dim = P.dimension()
        mult[dim] = mult.get(dim, 0) + 1
    return max(mult.values())
# Eric M. Schmidt, May 01 2013

More terms from Eric M. Schmidt, May 01 2013

A060426 a(n) is the number of degrees in the sequence of the degrees of the irreducible representations of the symmetric group S_n that appear only once.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 3, 4, 4, 5, 5, 4, 6, 8, 8, 6, 7, 10, 11, 11, 15, 15, 16, 18, 21, 22, 23, 29, 33, 31, 31, 39, 43, 44, 52, 51, 58, 64, 71, 66, 82, 88, 96, 93, 103, 115, 128, 143, 150, 156, 160, 173, 199, 202, 202, 242, 263, 269, 293, 308

Offset: 1

Avi Peretz (njk(AT)netvision.net.il), Apr 05 2001

nonn

Bounded above by A000700(n). - Eric M. Schmidt, Apr 29 2013

			a(6) = 1 because the degrees for S_6 are 1,1,5,5,5,5,9,9,10,10,16 and the only number that appears once is 16.

Cf. A059867, A060368, A060369, A060437, A061569, A089248. [From M. F. Hasler, Jun 14 2009]

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] = If[n == 1, 1, Count[Tally[g[n, n, {}]], {_, 1}] ];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 50}] (* Jean-François Alcover, Sep 23 2024, after Alois P. Heinz in A060240 *)

def A060426(n) :
    mult = {}
    for P in Partitions(n) :
        dim = P.dimension()
        mult[dim] = mult.get(dim, 0) + 1
    return len([m for m in iter(mult) if mult[m]==1])
# Eric M. Schmidt, Apr 29 2013

More terms from Eric M. Schmidt, Apr 29 2013

A117500 Triangle read by rows in which row n gives the partition of n associated with highest degree representation of symmetric group S_n.

Original entry on oeis.org

1, 2, 2, 1, 3, 1, 3, 1, 1, 3, 2, 1, 4, 2, 1, 4, 2, 1, 1, 4, 3, 1, 1, 4, 3, 2, 1, 5, 3, 2, 1, 5, 3, 2, 1, 1, 5, 4, 2, 1, 1, 6, 4, 2, 1, 1, 5, 4, 3, 2, 1, 6, 4, 3, 2, 1, 6, 4, 3, 2, 1, 1, 7, 4, 3, 2, 1, 1, 7, 5, 3, 2, 1, 1, 7, 5, 3, 2, 2, 1, 7, 5, 3, 2, 2, 1, 1, 7, 5, 4, 3, 2, 1, 7, 5, 4, 3, 2, 1

Offset: 1

N. J. A. Sloane, Apr 28 2006

Note that a partition and its conjugate give the same degree representation of the symmetric group. We take the lexicographically earlier of the two.

			Triangle begins:
1
2
2 1
3 1
3 1 1
3 2 1
4 2 1
4 2 1 1
4 3 1 1
4 3 2 1
5 3 2 1
5 3 2 1 1
5 4 2 1 1
6 4 2 1 1
5 4 3 2 1
6 4 3 2 1
6 4 3 2 1 1
7 4 3 2 1 1
7 5 3 2 1 1
7 5 3 2 2 1
7 5 3 2 2 1 1
7 5 4 3 2 1
7 5 4 3 2 1 1
8 5 4 3 2 1 1
8 6 4 3 2 1 1

J. McKay, The largest degrees of irreducible characters of the symmetric group. Math. Comp. 30 (1976), no. 135, 624-631. (Gives first 75 rows on pp. 627-629.)
J. McKay, Page 1 of 5 pages of tables from Math. Comp. paper
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

See A003040 for much more information. Cf. A060240.

If p_1 >= p_2 >= ... >= p_k is the partition of n, the degree of the representation (given in A003040) is n! * Product_{i

A224653 Irregular table which shows in row n the dimensions of the irreducible representations of the permutation group of order n!.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 3, 1, 4, 5, 6, 1, 5, 9, 10, 16, 1, 6, 14, 15, 20, 21, 35, 1, 7, 14, 20, 21, 28, 35, 42, 56, 64, 70, 90, 1, 8, 27, 28, 42, 48, 56, 70, 84, 105, 120, 162, 168, 189, 216, 1, 9, 35, 36, 42, 75, 84, 90, 126, 160, 210, 225, 252, 288, 300, 315, 350, 448, 450, 525, 567, 768

Offset: 0

R. J. Mathar, May 09 2013

This is triangle A060240 if duplicates in individual rows are removed.

The entries in row n give the number of standard Young tableaux of the Ferrers diagrams of the partitions of n (without duplicates, increasingly). Example: n = 4; there are 5 partitions of 4: [4], [3,1], [2,2], [2,1,1], and [1,1,1,1]; their Ferrers graphs have 1, 3, 2, 3, and 1 standard tableaux, respectively. - Emeric Deutsch, May 26 2015

			The group of permutations of [8] has 2 representations of dimension 1, 2 of dimension 7, 2 of dimension 14, 2 of dimension 20, 2 of dimension 21, 2 of dimension 28, 2 of dimension 35, 1 of dimension 42, 2 of dimension 56, 2 of dimension 64, 2 of dimension of 70 and 1 of dimension 90.
1;
1;
1;
1,2;
1,2,3;
1,4,5,6;
1,5,9,10,16;
1,6,14,15,20,21,35;
1,7,14,20,21,28,35,42,56,64,70,90;
1,8,27,28,42,48,56,70,84,105,120,162,168,189,216;
1,9,35,36,42,75,84,90,126,160,210,225,252,288,300,315,350,448,450,525,567,768;

Alois P. Heinz, Rows n = 0..40, flattened

Cf. A060240.

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))):
T:= n-> sort([{g(n$2, [])}[]])[]:
seq(T(n), n=0..12);  # Alois P. Heinz, May 26 2015

h[l_List] := Module[{n}, 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_List] := If[n==0 || i==1, h[Join[l, Array[1&, n]]], If[i<1, 0,Table[g[n - i*j, i-1, Join [l, Array[i&, j]]], {j, 0, n/i}]]]; T[n_] := g[n, n, {}] // Flatten // Union; T[1] = {1}; Table[T[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Jul 03 2015, after Alois P. Heinz *)

A027842 Number of subgroups of index n in fundamental group of a certain fiber space.

Original entry on oeis.org

1, 7, 34, 227, 1296, 10576, 77554, 729379, 7013419, 77371042, 910455096, 11877617828, 165087132666, 2480783846184, 39677415301224, 675818886754115, 12180498014887816, 231893906693650507, 4645789707491981892, 97740610362119850762, 2153888078209948673032

Offset: 1

N. J. A. Sloane

nonn

V. A. Liskovets and A. Mednykh, Enumeration of subgroups in the fundamental groups of orientable circle bundles over surfaces, Commun. in Algebra, 28, No. 4 (2000), 1717-1738.

Cf. A027838, A027839, A027840, A060240.

More terms from Sean A. Irvine, Dec 07 2019

A093716 Hook products of all partitions of 5.

Original entry on oeis.org

20, 24, 24, 30, 30, 120, 120

Offset: 1

Emeric Deutsch, May 17 2004

Cf. A003869, A060240.

Row n=5 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(5)[q])), q=1..numbpart(5))]);

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[5!/T[5]] (* Jean-François Alcover, Aug 12 2024, after Alois P. Heinz in A060240 *)

a(n) = 5!/A003869(8-n).

A093764 Hook products of all partitions of 6.

Original entry on oeis.org

45, 72, 72, 80, 80, 144, 144, 144, 144, 720, 720

Offset: 1

Emeric Deutsch, May 17 2004

Row n=6 of A093784.

Cf. A003870, A060240.

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(6)[q])),q=1..numbpart(6))]);

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[6!/T[6]] (* Jean-François Alcover, Jul 20 2024, after Alois P. Heinz in A060240 *)

a(n) = 6!/A003870(12-n).

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A060246 Triangle whose rows are the degrees of the irreducible representations of the groups PSL(2,p) as p runs through the primes.

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A089248 a(n) is the sum of the odd degrees of the irreducible representations of the symmetric group S_n.

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A104707 Triangle read by rows distributing the 1602 multinomials described by A005651(6) related to Young tableau and Kostka numbers.

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A060369 a(n) is the maximum number of occurrences of a degree in the sequence of the degrees of the irreducible representations of the symmetric group S_n.

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A060426 a(n) is the number of degrees in the sequence of the degrees of the irreducible representations of the symmetric group S_n that appear only once.

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A117500 Triangle read by rows in which row n gives the partition of n associated with highest degree representation of symmetric group S_n.

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A224653 Irregular table which shows in row n the dimensions of the irreducible representations of the permutation group of order n!.

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A027842 Number of subgroups of index n in fundamental group of a certain fiber space.

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