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 1-10 of 11 results. Next

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

A094152 a(n) is the position of prime 7 in the Euclid-Mullin (EM) sequence of type A000945, if it were started with prime(n) instead of 2.

Original entry on oeis.org

3, 3, 15, 1, 5, 6, 5, 24, 10, 6, 7, 6, 5, 4, 7, 5, 3, 5, 6, 16, 5, 6, 5, 28, 6, 3, 5, 36, 7, 15, 4, 15, 7, 7, 8, 7, 7, 5, 7, 14, 5, 6, 19, 16, 17, 5, 4, 12, 5, 8, 10, 17, 5, 5, 8, 10, 3, 5, 7, 30, 5, 5, 20, 3, 5, 6, 6, 4, 9, 9, 3, 9, 5, 6, 8, 8
Offset: 1

Views

Author

Labos Elemer, May 05 2004

Keywords

Examples

			n=8: p(8)=19, the corresponding EM sequence is A051312 in which p=7 arises at the 24th position as follows:
{19, 2, 3, 5, 571, 271, 457, 397, 1123, 23, 103, 42572757267735264511, 313, 17, 16013177, 7951, 1259, 41, 1531, 11, 83, 53, 67, 7, 21397}, thus a(8)=24.

Links

Crossrefs

Cf. A000945, A051308-A051334, A056756, A093777-A093784.

Extensions

More terms from Sean A. Irvine, Sep 20 2012

A093716 Hook products of all partitions of 5.

Original entry on oeis.org

20, 24, 24, 30, 30, 120, 120
Offset: 1

Views

Author

Emeric Deutsch, May 17 2004

Keywords

Crossrefs

Cf. A003869, A060240.
Row n=5 of A093784.

Programs

  • Maple
    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))]);
  • Mathematica
    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 *)

Formula

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

Views

Author

Emeric Deutsch, May 17 2004

Keywords

Crossrefs

Row n=6 of A093784.
Cf. A003870, A060240.

Programs

  • Maple
    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))]);
  • Mathematica
    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 *)

Formula

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

A093769 Hook products of all partitions of 7.

Original entry on oeis.org

144, 144, 240, 240, 252, 336, 336, 360, 360, 360, 360, 840, 840, 5040, 5040
Offset: 1

Views

Author

Emeric Deutsch, May 17 2004

Keywords

Crossrefs

Row n=7 of A093784.
Cf. A003871, A060240.

Programs

  • Maple
    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(7)[q])),q=1..numbpart(7))]);
  • Mathematica
    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[7!/T[7]] (* Jean-François Alcover, Aug 12 2024, after Alois P. Heinz in A060240 *)

Formula

a(n) = 7!/A003871(16-n).

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

Views

Author

Emeric Deutsch, May 17 2004

Keywords

Links

Crossrefs

Cf. A003876, A060240.
Row n=12 of A093784.

Programs

  • Maple
    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))]);
  • Mathematica
    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 *)

Formula

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

A093792 Hook products of all partitions of 13.

Original entry on oeis.org

290304, 290304, 302400, 302400, 362880, 362880, 388800, 414720, 414720, 483840, 483840, 518400, 518400, 518400, 518400, 537600, 537600, 544320, 544320, 604800, 604800, 665280, 665280, 691200, 691200, 725760, 725760, 725760, 798336, 798336, 844800, 844800, 907200
Offset: 1

Views

Author

Emeric Deutsch, May 17 2004

Keywords

Links

Crossrefs

Cf. A003877, A060240.
Row n=13 of A093784.

Programs

  • Maple
    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(13)[q])),q=1..numbpart(13))]);
  • Mathematica
    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[13!/T[13]] (* Jean-François Alcover, Sep 22 2024, after Alois P. Heinz in A060240 *)

Formula

a(n) = 13!/A003877(102-n).

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

Views

Author

Emeric Deutsch, May 17 2004

Keywords

Crossrefs

Cf. A003872, A060240.
Row n=8 of A093784.

Programs

  • Maple
    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))]);
  • Mathematica
    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 *)

Formula

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

A093787 Hook products of all partitions of 9.

Original entry on oeis.org

1680, 1680, 1920, 1920, 2160, 2160, 2240, 2240, 3024, 3024, 3456, 3456, 4320, 4320, 5184, 6480, 6480, 7560, 7560, 8640, 8640, 8640, 12960, 12960, 13440, 13440, 45360, 45360, 362880, 362880
Offset: 1

Views

Author

Emeric Deutsch, May 17 2004

Keywords

Crossrefs

Cf. A003873, A060240.
Row n=9 of A093784.

Programs

  • Maple
    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(9)[q])),q=1..numbpart(9))]);
  • Mathematica
    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[9!/T[9]] (* Jean-François Alcover, Aug 12 2024, after Alois P. Heinz in A060240 *)

Formula

a(n) = 9!/A003873(31-n).

A093789 Hook products of all partitions of 10.

Original entry on oeis.org

4725, 6400, 6400, 6912, 6912, 8064, 8064, 8100, 10368, 10368, 11520, 11520, 12096, 12096, 12600, 12600, 14400, 14400, 16128, 16128, 17280, 17280, 22680, 22680, 28800, 28800, 40320, 40320, 43200, 43200, 48384, 48384, 86400, 86400, 100800, 100800, 103680, 103680, 403200, 403200, 3628800, 3628800
Offset: 1

Views

Author

Emeric Deutsch, May 17 2004

Keywords

Comments

All 42 terms of this finite sequence are shown.

Crossrefs

Cf. A003874, A060240.
Row n=10 of A093784.

Programs

  • Maple
    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(10)[q])),q=1..numbpart(10))]);
  • Mathematica
    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[10!/T[10]] (* Jean-François Alcover, Aug 12 2024, after Alois P. Heinz in A060240 *)

Formula

a(n) = 10!/A003874(43-n).
Showing 1-10 of 11 results. Next

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

Content is available under The OEIS End-User License Agreement.