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.

A213322 Number of permutations of n objects such that no three-element subset is preserved.

Original entry on oeis.org

1, 1, 2, 0, 9, 54, 459, 2568, 20145, 176076, 1833741, 20148336, 241870617, 3132196560, 43874128089, 658195206264, 10533823597089, 179062417518768, 3223079582143185, 61237777946016096, 1224762717659002281, 25720036368344942616, 565841009719801635777
Offset: 0

Views

Author

Les Reid, Jun 08 2012

Keywords

Comments

The limit as n -> infinity of a(n)/n! = (3+2*exp(1/2))/(2*exp(11/6)) or approximately 0.5034167572.

Examples

			Example: For n=5 the only permutations that fix no three-element subset are the 24 5-cycles and the 30 4-cycles, therefore a(5)=54.

Crossrefs

Cf. A000166, A137482, A213323, A213324.

Programs

  • PARI
    lista(nn) = {x=xx+O(xx^nn); egf=((x+x^2/2)*exp(-x-x^2/2-x^3/3)+exp(-x-x^3/3))/(1-x); Vec(serlaplace(egf)) ;} \\ Michel Marcus, Aug 14 2013

Formula

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

Extensions

More terms from Michel Marcus, Aug 14 2013

A213323 Number of permutations of n objects such that no four-element subset is preserved.

Original entry on oeis.org

1, 1, 2, 6, 0, 44, 304, 2568, 26704, 200240, 1931616, 20849696, 246556672, 3300906816, 46382446720, 695413794944, 11120648673024, 188600719094528, 3394592207824384, 64513420630110720, 1290420198709682176, 27102196040419214336, 596237419436696543232, 13713106494042086045696
Offset: 0

Views

Author

Les Reid, Jun 08 2012

Keywords

Comments

The limit as n -> infinity of a(n)/n! = (13+9*exp(1/3))/(6*exp(25/12)) or approximately 0.5304422700.

Examples

			Example: For n=5 the only permutations that fix no four-element subset are the 24 5-cycles and the 20 products of a 3-cycle and a 2-cycle, therefore a(5)=44.

Crossrefs

Cf. A000166, A137482, A213322, A213324

Programs

  • PARI
    x='x+O('x^66);
egf=((x+x^2/2+2*x^3/3)*exp(-x-x^2/2-x^3/3-x^4/4)+(1+x^2/2)*exp(-x-x^2/2-x^4/4))/(1-x);
Vec(serlaplace(egf))
/* Joerg Arndt, Jun 09 2012 */

Formula

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

A213324 Number of permutations of n objects such that no five-element subset is preserved.

Original entry on oeis.org

1, 1, 2, 6, 24, 0, 265, 2260, 20145, 200240, 2492225, 23163480, 270877705, 3449462080, 48030998625, 713129276000, 11685451112225, 198919432944000, 3585292622812225, 68053546078588000, 1360638669122771625, 28525836193802883200, 627637954389517169825, 14435957818250131813200, 346518764145610187160625
Offset: 0

Views

Author

Les Reid, Jun 08 2012

Keywords

Comments

Limit_{n->oo} a(n)/n! = (35-24*exp(1/4)+24*exp(1/3)+24*exp(7/12)+24*exp(3/4))/(24*exp(137/60)) = 0.5585422951...

Examples

			For n=6 the only permutations that fix no five-element subset are the 120 6-cycles, the 90 products of a 4-cycle and a 2-cycle, the 40 products of two 3-cycles, and the 15 products of three 2-cycles, therefore a(5)=265.

Crossrefs

Cf. A000166, A137482, A213322, A213323

Programs

  • PARI
    x='x+O('x^66);
egf=((x^2/2+2*x^3/3+7*x^4/24)*exp(-x-x^2/2-x^3/3-x^4/4-x^5/5)+x*exp(-x-x^2/2-x^4/4-x^5/5)+exp(-x-x^2/2-x^5/5)+exp(-x-x^3/3-x^5/5)-exp(-x-x^2/2-x^3/3-x^5/5))/(1-x);
Vec(serlaplace(egf))
/* Joerg Arndt, Jun 09 2012 */

Formula

E.g.f.: ((x^2/2+2*x^3/3+7*x^4/24)*exp(-x-x^2/2-x^3/3-x^4/4-x^5/5)+x*exp(-x-x^2/2-x^4/4-x^5/5)+exp(-x-x^2/2-x^5/5)+exp(-x-x^3/3-x^5/5)-exp(-x-x^2/2-x^3/3-x^5/5))/(1-x).

A344901 Triangle read by rows: T(n,k) is the number of permutations of length n that have k same elements at the same positions with its inverse permutation for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 0, 2, 2, 0, 0, 4, 6, 8, 0, 0, 10, 24, 30, 40, 0, 0, 26, 160, 144, 180, 160, 0, 0, 76, 1140, 1120, 1008, 840, 700, 0, 0, 232, 8988, 9120, 8960, 5376, 4200, 2912, 0, 0, 764, 80864, 80892, 82080, 53760, 30240, 19656, 12768, 0, 0, 2620, 809856, 808640, 808920, 547200, 336000, 157248, 95760, 55680, 0, 0, 9496
Offset: 0

Views

Author

Mikhail Kurkov, Jun 01 2021

Keywords

Examples

			Triangle T(n,k) begins:
     1;
     0,    1;
     0,    0,    2;
     2,    0,    0,    4;
     6,    8,    0,    0,   10;
    24,   30,   40,    0,    0,   26;
   160,  144,  180,  160,    0,    0, 76;
  1140, 1120, 1008,  840,  700,    0,  0, 232;
  8988, 9120, 8960, 5376, 4200, 2912,  0,   0, 764;
  ...

Links

Crossrefs

Columns k=0-1 give: A038205, A221145.
Row sums give A000142.
Main diagonal gives A000085.
Cf. A000023, A007318, A052571, A052849, A098558, A137482.

Programs

  • Maple
    b:= proc(n, t) option remember; `if`(n=0, 1, add(b(n-j, t)*
      binomial(n-1, j-1)*(j-1)!, j=`if`(t=1, 1..min(2, n), 3..n)))
    end:
T:= (n, k)-> binomial(n, k)*b(k, 1)*b(n-k, 0):
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Oct 28 2024
  • Mathematica
    b[n_, t_] := b[n, t] = If[n == 0, 1, Sum[b[n-j, t]* Binomial[n-1, j-1]*(j-1)!, {j, If[t == 1, Range @ Min[2, n], Range[3, n]]}]];
T[n_, k_] := Binomial[n, k]*b[k, 1]*b[n-k, 0];
Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* Jean-François Alcover, Apr 24 2025, after Alois P. Heinz *)

Formula

T(n,k) = binomial(n,k)*A000085(k)*A038205(n-k).
From Alois P. Heinz, Oct 28 2024: (Start)
Sum_{k=0..n} k * T(n,k) = A052849(n) = A098558(n) for n>=2.
Sum_{k=0..n} (n-k) * T(n,k) = A052571(n).
Sum_{k=0..n} (-1)^k * T(n,k) = A000023(n).
T(n,0) + T(n,1) = A137482(n). (End)

A306191 T(n,k) is a triangular array read by rows. Let S_n act on the set of size two subsets of {1,2,...,n}. T(n,k) is the number of permutations in S_n that fix exactly k size two subsets, n >= 1, 0 <= k <= binomial(n,2).

Original entry on oeis.org

1, 0, 2, 2, 3, 0, 1, 14, 0, 9, 0, 0, 0, 1, 54, 40, 15, 0, 10, 0, 0, 0, 0, 0, 1, 304, 300, 0, 100, 0, 0, 0, 15, 0, 0, 0, 0, 0, 0, 0, 1, 2260, 1638, 630, 315, 0, 105, 70, 0, 0, 0, 0, 21, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 18108, 12992, 5460, 1344, 1645, 0, 420, 0, 210, 0, 112, 0, 0, 0, 0, 0, 28, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
Offset: 1

Views

Author

Geoffrey Critzer, Jan 28 2019

Keywords

Comments

The action of S_n on the 2-subsets of {1,2,...,n} is defined: For all pi in S_n, pi({i,j}) = {pi(i),pi(j)}.

Examples

			1,
0,   2,
2,   3,   0,  1,
14,  0,   9,  0,   0,  0, 1,
54,  40,  15, 0,   10, 0, 0, 0,  0, 0, 1,
304, 300, 0,  100, 0,  0, 0, 15, 0, 0, 0, 0, 0, 0, 0, 1,

Crossrefs

Cf. A137482 is column 1.

Programs

  • Mathematica
    f[list_] := Flatten[Position[list /. x_ /; x > 0 -> 1, 1]];
Level[CoefficientList[Table[n! PairGroupIndex[SymmetricGroup[n], s] /. {Table[s[i] -> 1, {i, 2, Binomial[n, 2]}]}, {n, 1, 8}],
   s[1]], {2}] // Grid
Showing 1-5 of 5 results.

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