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 31-39 of 39 results.

A243748 Irregular triangle read by rows where T(n,k) is the number of subgroups of order d of the symmetric group S_n, where d is the k-th divisor of n!.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 1, 9, 4, 7, 4, 3, 1, 1, 1, 25, 10, 35, 6, 30, 15, 6, 15, 0, 6, 5, 0, 0, 1, 1, 1, 75, 40, 255, 36, 280, 255, 10, 36, 150, 0, 45, 50, 36, 90, 0, 30, 0, 0, 30, 12, 10, 0, 0, 12, 0, 0, 0, 1, 1, 1, 231, 175, 1295, 126, 1645, 120, 1575, 70, 378, 1715, 120, 0, 315, 350, 378, 120, 1435, 0, 0, 0, 245, 126, 120, 0
Offset: 1

Views

Author

R. J. Mathar, Jun 09 2014

Keywords

Comments

The columns skip counting the subgroups of S_n with orders d that do not divide the order of S_n, n!, because such subgroups cannot exist. This is just a reduction of columns in the triangle by omitting a large number of zeros.

Examples

			There are T(3,2)=3 subgroups of S_3 of order 2, namely the groups generated by the permutations (1,2), (1,3) or (2,3).
Triangle begins:
  1;
  1,1;
  1,3,1,1;
  1,9,4,7,4,3,1,1;
  1,25,10,35,6,30,15,6,15,0,6,5,0,0,1,1;
  ...

Crossrefs

Cf. A005432 (row sums), A001189 (column d=2), A027423 (row lengths), A218913, A277566, A284210.

Programs

  • GAP
    # GAP 4
LoadPackage("SONATA") ;;
Print("\n") ;
N := Factorial(7) ;; # adjusted to the maximum n below
subS := EmptyPlist(N) ;;
for n in [1..7] do
    for e in [1..N] do
        subS[e] := 0 ;
    od;
    g := SymmetricGroup(n) ;
    sg := Size(g) ;
    alls := Subgroups(g) ;
    for s in alls do
        o := Size(s) ;
        if o <= N then
            subS[o] := subS[o]+1 ;;
        fi;
    od ;
    for d in [1..N] do
        if ( sg mod d ) = 0 then
            Print(subS[d],",") ;
        fi;
    od;
    Print("\n") ;
od;

Extensions

Edited by Peter Munn, Mar 06 2025

A061138 Number of degree-n odd permutations of order exactly 4.

Original entry on oeis.org

0, 0, 0, 0, 6, 30, 90, 210, 1680, 12096, 114660, 833580, 5928120, 38112360, 259194936, 1739195640, 17043237120, 167089937280, 1837707369840, 18342985021776, 181206905922720, 1673742164139360, 16992525855006240
Offset: 0

Views

Author

Vladeta Jovovic, Apr 14 2001

Keywords

Crossrefs

Cf. A000085, A001470, A001472, A052501, A053496 - A053505, A001189, A001471, A001473, A061121 - A061128, A000704, A061129 - A061132, A048099, A051695, A061133 - A061135, A001465, A061136 - A061140.

Formula

E.g.f.: - 1/2*exp(x + 1/2*x^2) + 1/2*exp(x - 1/2*x^2) + 1/2*exp(x + 1/2*x^2 + 1/4*x^4) - 1/2*exp(x - 1/2*x^2 - 1/4*x^4).

A061139 Number of degree-n odd permutations of order exactly 6.

Original entry on oeis.org

0, 0, 0, 0, 0, 20, 240, 1260, 5600, 45360, 383040, 2451680, 17128320, 157769040, 1902380480, 18882623760, 163633317120, 2095059774080, 30792478993920, 346562329685760, 3905491275514880, 58609449249207360, 866031730098205440
Offset: 0

Views

Author

Vladeta Jovovic, Apr 14 2001

Keywords

Crossrefs

Cf. A000085, A001470, A001472, A052501, A053496 - A053505, A001189, A001471, A001473, A061121 - A061128, A000704, A061129 - A061132, A048099, A051695, A061133 - A061135, A001465, A061136 - A061140.

Formula

E.g.f.: - 1/2*exp(x + 1/2*x^2) + 1/2*exp(x - 1/2*x^2) + 1/2*exp(x + 1/2*x^2 + 1/3*x^3 + 1/6*x^6) - 1/2*exp(x - 1/2*x^2 + 1/3*x^3 - 1/6*x^6).

A061122 Number of degree-n permutations of order exactly 8.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 5040, 45360, 453600, 3326400, 39916800, 363242880, 3874590720, 34767532800, 567177811200, 6897521030400, 98241008785920, 1138935652807680, 18952720774041600, 258251731634534400
Offset: 1

Views

Author

Vladeta Jovovic, Apr 14 2001

Keywords

Crossrefs

Cf. A000085, A001470, A001472, A052501, A053496-A053505, A001189, A001471, A001473, A061121-A061128.

Formula

E.g.f.: -exp(x+1/2*x^2+1/4*x^4)+exp(x+1/2*x^2+1/4*x^4+1/8*x^8).

A061123 Number of degree-n permutations of order exactly 9.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 40320, 403200, 2217600, 26611200, 259459200, 1695133440, 16345929600, 161902540800, 1208560953600, 50830132953600, 866513503215360, 8470676211379200, 166891791625977600, 2699606616475507200
Offset: 1

Views

Author

Vladeta Jovovic, Apr 14 2001

Keywords

Crossrefs

Cf. A000085, A001470, A001472, A052501, A053496-A053505, A001189, A001471, A001473, A061121-A061128.

Formula

E.g.f.: -exp(x+1/3*x^3)+exp(x+1/3*x^3+1/9*x^9).

A061124 Number of degree-n permutations of order exactly 10.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 504, 4032, 27216, 514080, 4823280, 57081024, 500972472, 4412103696, 60619398840, 686638592640, 9335025764064, 104304736815552, 1180585704051936, 29016515871665280, 478096386437121480
Offset: 1

Views

Author

Vladeta Jovovic, Apr 14 2001

Keywords

Crossrefs

Cf. A000085, A001470, A001472, A052501, A053496-A053505, A001189, A001471, A001473, A061121-A061128.

Formula

E.g.f.: exp(x) - exp(x+1/2*x^2) - exp(x+1/5*x^5) + exp(x+1/2*x^2+1/5*x^5+1/10*x^10).
From Benedict W. J. Irwin, May 27 2016: (Start)
Let y1(0)=1, y1(1)=1,
Let -y1(n)-y1(n+1)+(n+2)*y1(n+2)=0,
Let y2(0)=1, y2(1)=1, y2(2)=1/2, y2(3)=1/6, y2(4)=1/24,
Let -y2(n)-y2(n+4)+(n+5)*y2(n+5)=0,
Let y3(0)=1, y3(1)=1, y3(2)=1, y3(3)=2/3, y3(4)=5/12, y3(5)=5/12, y3(6)=11/36, y3(7)=31/126, y3(8)=307/2016, y3(9)=1643/18144,
Let -y3(n)-y3(n+5)-y3(n+8)-y3(n+9)+(n+10)*y3(n+10)=0,
a(n) = 1+n!*(y3(n)-y2(n)-y1(n)).
(End)

A061125 Number of degree-n permutations of order exactly 12.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 420, 3360, 30240, 403200, 4019400, 80166240, 965284320, 12173441280, 162850287600, 2428557331200, 32123543612160, 534678700308480, 8126981741380320, 128338880777251200, 2080312367956502400, 36351373041072122880, 606331931399062693440
Offset: 1

Views

Author

Vladeta Jovovic, Apr 14 2001

Keywords

Crossrefs

Cf. A000085, A001470, A001472, A052501, A053496 - A053505, A001189, A001471, A001473, A061121 - A061128.

Programs

  • Mathematica
    nn=21;Range[0,nn]!CoefficientList[Series[(Exp[x^12/12]-1)Exp[x+x^2/2+x^3/3+x^4/4+x^6/6]+(Exp[x^6/6]-1)(Exp[x^4/4]-1)Exp[x+x^2/2+x^3/3]+(Exp[x^4/4]-1)(Exp[x^3/3]-1)Exp[x^2/2+x],{x,0,nn}],x]//Rest  (* Geoffrey Critzer, Feb 04 2013 *)

Formula

E.g.f.: exp(x + 1/2*x^2) - exp(x + 1/2*x^2 + 1/4*x^4) - exp(x + 1/2*x^2 + 1/3*x^3 + 1/6*x^6) + exp(x + 1/2*x^2 + 1/3*x^3 + 1/4*x^4 + 1/6*x^6 + 1/12*x^12).

A061126 Number of degree-n permutations of order exactly 16.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1307674368000, 22230464256000, 400148356608000, 5068545850368000, 101370917007360000, 1490152480008192000, 24977793950613504000, 343667682838351872000
Offset: 1

Views

Author

Vladeta Jovovic, Apr 14 2001

Keywords

Crossrefs

Cf. A000085, A001470, A001472, A052501, A053496 - A053505, A001189, A001471, A001473, A061121 - A061128.

Formula

E.g.f.: - exp(x + 1/2*x^2 + 1/4*x^4 + 1/8*x^8) + exp(x + 1/2*x^2 + 1/4*x^4 + 1/8*x^8 + 1/16*x^16).

A061127 Number of degree-n permutations of order exactly 24.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1663200, 19958400, 259459200, 4843238400, 72648576000, 988020633600, 14600749363200, 224704121241600, 3614691131251200, 84808750650624000, 1509309706083379200, 29359195162807910400
Offset: 1

Views

Author

Vladeta Jovovic, Apr 14 2001

Keywords

Crossrefs

Cf. A000085, A001470, A001472, A052501, A053496 - A053505, A001189, A001471, A001473, A061121 - A061128.

Programs

  • Mathematica
    nn=22;Range[0,nn]!CoefficientList[Series[(Exp[x^24/24]-1)Exp[x+x^2/2+x^3/3+x^4/4+x^6/6+x^8/8+x^12/12]+(Exp[x^12/12]-1)(Exp[x^8/8]-1)Exp[x+x^2/2+x^3/3+x^4/4+x^6/6]+(Exp[x^8/8]-1)(Exp[x^6/6]-1)Exp[x+x^2/2+x^3/3+x^4/4]+(Exp[x^8/8]-1)(Exp[x^3/3]-1)Exp[x+x^2/2+x^4/4],{x,0,nn}],x]//Rest (* Geoffrey Critzer, Feb 04 2013 *)

Formula

E.g.f.: exp(x + 1/2*x^2 + 1/4*x^4) - exp(x + 1/2*x^2 + 1/4*x^4 + 1/8*x^8) - exp(x + 1/2*x^2 + 1/3*x^3 + 1/4*x^4 + 1/6*x^6 + 1/12*x^12) + exp(x + 1/2*x^2 + 1/3*x^3 + 1/4*x^4 + 1/6*x^6 + 1/8*x^8 + 1/12*x^12 + 1/24*x^24).
Previous Showing 31-39 of 39 results.

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