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 1-3 of 3 results.

A144085 a(n) is the number of partial bijections (or subpermutations) of an n-element set without fixed points (also called partial derangements).

Original entry on oeis.org

1, 1, 4, 18, 108, 780, 6600, 63840, 693840, 8361360, 110557440, 1590351840, 24713156160, 412393101120, 7352537512320, 139443752448000, 2802408959750400, 59479486120454400, 1329239028813696000, 31194214921732262400, 766888191387539020800, 19707387644116280908800, 528327710066244459571200
Offset: 0

Views

Author

Abdullahi Umar, Sep 10 2008, Sep 15 2008

Keywords

Comments

a(n) is also the number of matchings on the n-crown graph. - Eric W. Weisstein, Jul 11 2011

Examples

			a(3) = 18 because there are exactly 18 partial derangements (on a 3-element set), namely: the empty map, (1)->(2), (1)->(3), (2)->(1), (2)->(3), (3)->(1), (3)->(2), (1,2)->(2,1), (1,2)->(2,3), (1,2)->(3,1), (1,3)->(2,1), (1,3)->(3,1), (1,3)->(3,2), (2,3)->(1,2), (2,3)->(3,1), (2,3)->(3,2), (1,2,3)->(2,3,1), (1,2,3)->(3,1,2) - the mappings are coordinate-wise.

Links

Crossrefs

Cf. A144088.
Column k=0 of A369292.

Programs

  • Maple
    A144085 := proc(n)
    option remember;
    if n < 0 then
        0 ;
    elif n < 2 then
        1;
    else
        (2*n-1)*procname(n-1)-(n-1)*(n-3)*procname(n-2)-(n-1)*(n-2)*procname(n-3) ;
    end if;
end proc: # R. J. Mathar, Nov 03 2015
  • Mathematica
    Table[n! Sum[(-1)^k/k! LaguerreL[n - k, -1], {k, 0, n}], {n, 0, 30}]
RecurrenceTable[{n (1 + n) a[n] + (-1 + n^2) a[1 + n] + a[3 + n] == (3 + 2 n) a[2 + n], a[1] == 1, a[2] == 1, a[3] == 4}, a, {n, 20}] (* Eric W. Weisstein, Sep 30 2017 *)
  • PARI
    x='x+O('x^66);
k=0; egf=x^k/k!*exp(x^2/(1-x))/(1-x);
Vec(serlaplace(egf)) /* Joerg Arndt, Jul 11 2011 */

Formula

a(n) = A144088(n,0).
a(n) = n! * Sum_{m=0..n} (-1^m/m!) * Sum_{j=0..n-m} binomial(n-m, j)/j!.
a(n) = (2*n-1)*a(n-1) - (n-1)*(n-3)*a(n-2) - (n-1)*(n-2)*a(n-3), a(0)=1, a(n)=0 if n < 0.
E.g.f. for number of partial bijections of an n-element set with exactly k fixed points is (x^k/k!)*exp(x^2/(1-x))/(1-x). - Vladeta Jovovic, Nov 09 2008
a(n) ~ exp(2*sqrt(n)-n-3/2)*n^(n+1/4)/sqrt(2) * (1+79/(48*sqrt(n))). - Vaclav Kotesovec, Aug 11 2013
a(n) = n! * Sum_{k=0..n} binomial(k,n-k)/(n-k)!. - Seiichi Manyama, Aug 06 2024

A144087 a(n) is the number of partial bijections (or subpermutations) of an n-element set with exactly 2 fixed points.

Original entry on oeis.org

0, 0, 1, 3, 24, 180, 1620, 16380, 184800, 2298240, 31222800, 459874800, 7296791040, 124047443520, 2248897210560, 43301275617600, 882304501478400, 18964350332928000, 428768570841811200, 10170992126597702400, 252555415474602240000, 6550785133563775104000, 177151172210521513804800
Offset: 0

Views

Author

Abdullahi Umar, Sep 11 2008

Keywords

Examples

			a(3) = 3 because there are exactly 3 partial bijections (on a 3-element set) with exactly 2 fixed points, namely: (1,2)->(1,2), (1,3)->(1,3), (2,3)->(2,3) - the mappings are coordinate-wise.

Links

Crossrefs

Column k=2 of A144088.
Cf. A144085.

Programs

  • PARI
    x='x+O('x^66); /* that many terms */
k=2; egf=x^k/k!*exp(x^2/(1-x))/(1-x);
Vec(serlaplace(egf)) /* show terms, starting with 1 */
/* Joerg Arndt, Jul 11 2011 */

Formula

a(n) = (n*(n-1)/2)*A144085(n-2).
E.g.f.: (x^k/k!)*exp(x^2/(1-x))/(1-x) where k=2. - Joerg Arndt, Jul 11 2011
a(n) = (n!/2)*Sum_{m=0..n-2} (-1^m/m!)*Sum_{j=0..n-m} C(n-m,j)/j!;
(n-2)*a(n) = n*(2*n-5)*a(n-1) - n*(n-1)*(n-5)*a(n-2) - n*(n-1)*(n-2)*a(n-3), a(2)=1 and a(n)=0 if n < 2.
a(n) ~ n^(n + 1/4) * exp(2*sqrt(n) - 3/2 - n) / 2^(3/2) * (1 - 17/(48*sqrt(n))). - Vaclav Kotesovec, Dec 01 2021
a(n) = (n!/2) * Sum_{k=0..n-2} binomial(k,n-2-k)/(n-2-k)!. - Seiichi Manyama, Aug 06 2024

A144086 Number of partial bijections (or subpermutations) of an n-element set with exactly 1 fixed point.

Original entry on oeis.org

0, 1, 2, 12, 72, 540, 4680, 46200, 510720, 6244560, 83613600, 1216131840, 19084222080, 321271030080, 5773503415680, 110288062684800, 2231100039168000, 47640952315756800, 1070630750168179200, 25255541547460224000, 623884298434645248000, 16104652019138319436800
Offset: 0

Views

Author

Abdullahi Umar, Sep 10 2008, Sep 15 2008

Keywords

Examples

			a(3) = 12 because there are exactly 12 partial bijections (on a 3-element set) with exactly 1 fixed point, namely: (1)->(1), (2)->(2), (3)->(3), (1,2)->(1,3), (1,2)->(3,2), (1,3)->(1,2), (1,3)->(2,3), (2,3)->(2,1), (2,3)->(1,3), (1,2,3)->(1,3,2), (1,2,3)->(3,2,1), (1,2,3)->(2,1,3) - the mappings are coordinate-wise.

Links

Crossrefs

Column k=1 of A144088.
Cf. A144085.

Programs

  • Mathematica
    CoefficientList[Series[x*E^(x^2/(1-x))/(1-x), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Feb 24 2014 *)
  • PARI
    x='x+O('x^66); /* that many terms */
k=1; egf=x^k/k!*exp(x^2/(1-x))/(1-x);
Vec(serlaplace(egf)) /* show terms, starting with 1 */
/* Joerg Arndt, Jul 11 2011 */

Formula

a(n) = n*A144085(n-1).
E.g.f.: (x^k/k!)*exp(x^2/(1-x))/(1-x) where k=1. - Joerg Arndt, Jul 11 2011
a(n) = n!*Sum_{m=0..n-1} (-1^m/m!)*Sum_{j=0..n-m} C(n-m)/j!;
(n-1)*a(n) = n*(2*n-3)*a(n-1) - n*(n-1)*(n-4)*a(n-2) - n*(n-1)*(n-2)*a(n-3), a(1)=1 and a(n)=0 if n < 1.
a(n) ~ n^(n+1/4) * exp(2*sqrt(n)-n-3/2) / sqrt(2) * (1 + 31/(48*sqrt(n))). - Vaclav Kotesovec, Feb 24 2014
a(n) = n! * Sum_{k=0..n-1} binomial(k,n-1-k)/(n-1-k)!. - Seiichi Manyama, Aug 06 2024
Showing 1-3 of 3 results.

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