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.

Previous Showing 31-38 of 38 results.

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).

A093790 Hook products of all partitions of 11.

Original entry on oeis.org

17280, 17280, 25920, 25920, 30240, 30240, 32400, 32400, 33600, 34560, 34560, 36288, 36288, 40320, 40320, 40320, 40320, 43200, 43200, 48384, 48384, 57600, 57600, 60480, 60480, 67200, 67200, 72576, 72576, 86400, 86400, 103680, 103680, 120960, 120960, 158400, 172800, 172800, 190080, 190080, 241920, 241920, 302400, 302400, 332640, 332640, 362880, 362880, 887040, 887040, 907200, 907200, 3991680, 3991680, 39916800, 39916800
Offset: 1

Views

Author

Emeric Deutsch, May 17 2004

Keywords

Comments

All 56 terms of this finite sequence are shown.

Crossrefs

Cf. A003875, A060240.
Row n=11 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(11)[q])), q=1..numbpart(11))]);
  • 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[11!/T[11]] (* Jean-François Alcover, Sep 05 2024, after Alois P. Heinz in A060240 *)

Formula

a(n) = 11!/A003875(57-n).

A318558 Number of degrees of irreducible representations of symmetric group S_n that appear more than once.

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 4, 6, 10, 14, 20, 26, 35, 43, 49, 77, 103, 125, 174, 190, 274, 340, 430, 496, 686, 838, 1026, 1263, 1579, 1832, 2457, 2833, 3631, 4249, 5114, 6111, 7962, 9072, 11015, 12939, 16173, 18304, 23101, 26188, 31822, 37518, 45073, 51403, 63489, 71822
Offset: 0

Views

Author

Pierandrea Formusa, Aug 28 2018

Keywords

Examples

			Number 4 has the following partitions: a) [4], b) [3, 1], c) [2, 2], d) [2, 1, 1], e) [1, 1, 1, 1]. For partition a the cardinality of standard Young tableaux is 1, for b 3, for c 2, for d 3 and for e 1, so multiple cardinalities are 1 and 3: two multiple cardinalities, i.e., 4th sequence element is 2.

Crossrefs

Cf. A060240, A060426, A060437.

Programs

  • 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}]]];
a[n_] := a[n] = If[n == 0 || n == 1, 0, Count[Tally[g[n, n, {}]], {, k /; k > 1}] ];
Table[Print[n, " ", a[n]]; a[n], {n, 0, 49}] (* Jean-François Alcover, Sep 23 2024, after Alois P. Heinz in A060240 *)
  • SageMath
    r=""
lista=[]
lista_rip=[]
rip=0
for i in range(1,35):
        l=Partitions(i)
        for p in l:
            nsc=StandardTableaux(p).cardinality()
            if nsc in lista:
                if nsc not in lista_rip:
                    lista_rip.append(nsc)
                    rip += 1
            else:
                lista.append(nsc)
        r = r+","+str(rip)
        rip=0
        lista=[]
        lista_rip=[]
print(r)

Formula

a(n) = A060437(n) - A060426(n). - Alois P. Heinz, Aug 29 2018

Extensions

a(42)-a(49) from Alois P. Heinz, Aug 29 2018
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