cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A060240 Triangle T(n,k) in which n-th row gives degrees of irreducible representations of symmetric group S_n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 3, 1, 1, 4, 4, 5, 5, 6, 1, 1, 5, 5, 5, 5, 9, 9, 10, 10, 16, 1, 1, 6, 6, 14, 14, 14, 14, 15, 15, 20, 21, 21, 35, 35, 1, 1, 7, 7, 14, 14, 20, 20, 21, 21, 28, 28, 35, 35, 42, 56, 56, 64, 64, 70, 70, 90, 1, 1, 8, 8, 27, 27, 28, 28, 42, 42, 42, 48, 48, 56, 56, 70, 84
Offset: 0

Views

Author

N. J. A. Sloane, Mar 21 2001

Keywords

Comments

Sum_{k>=1} T(n,k)^2 = n!. - R. J. Mathar, May 09 2013
From Emeric Deutsch, Oct 31 2014: (Start)
Number of entries in row n = A000041(n) = number of partitions of n.
Sum of entries in row n = A000085(n).
Largest (= last) entry in row n = A003040(n).
The entries in row n give the number of standard Young tableaux of the Ferrers diagrams of the partitions of n (nondecreasingly). (End)

Examples

			Triangle begins:
  1;
  1;
  1, 1;
  1, 1, 2;
  1, 1, 2, 3, 3;
  1, 1, 4, 4, 5, 5, 6;
  ...

References

  • J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups, Oxford Univ. Press, 1985.
  • B. E. Sagan, The Symmetric Group, 2nd ed., Springer, 2001, New York.

Links

Crossrefs

Rows give A003870, A003871, etc. Cf. A060241, A060246, A060247.
Maximal entry in each row gives A003040.
Cf. A000041, A000085, A000142, A060437, A093784, A224653.

Programs

  • Magma
    CharacterTable(SymmetricGroup(6)); // (say)
  • Maple
    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, n, [])])[]:
seq(T(n), n=0..10);  # Alois P. Heinz, Jan 07 2013
  • Mathematica
    h[l_List] := 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_List] := 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_] := Sort[g[n, n, {}]]; T[1] = {1};
Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Jan 27 2014, after Alois P. Heinz *)

Extensions

More terms from Vladeta Jovovic, May 20 2003

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

Views

Author

N. J. A. Sloane, Mar 22 2001

Keywords

Examples

			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, ...).

References

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

Links

Crossrefs

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.

Programs

  • Magma
    CharacterTable(PSL(2,7)); // (say)
  • Magma
    &cat[[Degree(irred): irred in CharacterTable(PSL(2,q))]: q in [2..17]| IsPrimePower(q)]; // Jason Kimberley, May 22 2010

Extensions

Extended by Jason Kimberley, May 22 2010

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

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane, Mar 22 2001

Keywords

Examples

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

References

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

Links

Crossrefs

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.

Programs

  • Magma
    CharacterTable(PSL(2,7)); (say)
  • Magma
    &cat[[Degree(irred): irred in CharacterTable(PSL(2, p))]: p in PrimesUpTo(30)];

Extensions

Extended by Jason Kimberley, May 23 2010

A003862 Degrees of irreducible representations of alternating group A_7.

Original entry on oeis.org

1, 6, 10, 10, 14, 14, 15, 21, 35
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • 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].

Crossrefs

A row of A060241.

Programs

  • GAP
    List(Irr(CharacterTable("A7")), chi->chi[1]); # Eric M. Schmidt, Jul 18 2012

A003863 Degrees of irreducible representations of alternating group A_8.

Original entry on oeis.org

1, 7, 14, 20, 21, 21, 21, 28, 35, 45, 45, 56, 64, 70
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

References

  • 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].

Crossrefs

A row of A060241.

Programs

  • GAP
    List(Irr(CharacterTable("A8")), chi->chi[1]); # Eric M. Schmidt, Jul 18 2012
Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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