A116218 -id:A116218 - cp's OEIS frontend

A374980 Number of multiset permutations of {1, 1, 2, 2, ..., n, n} with no fixed pair (j,j).

1, 0, 5, 74, 2193, 101644, 6840085, 630985830, 76484389121, 11792973495032, 2254432154097861, 523368281765512930, 145044815855963403985, 47302856057098946329284, 17933275902554972391519893, 7820842217155394547769452734, 3887745712142302082441578104705

Offset: 0

Alois P. Heinz, Aug 05 2024

nonn

Inverse binomial transform of A000680.

			a(2) = 5: 1212, 1221, 2112, 2121, 2211.

Alois P. Heinz, Table of n, a(n) for n = 0..238

Cf. A000166, A000459, A000680, A059841, A116218, A375222, A375223.

Column k=2 of A375694.

a:= proc(n) option remember; `if`(n<3, [1, 0, 5][n+1],
     (n-1)*((2*n+1)*a(n-1)+(4*n-3)*a(n-2)+2*(n-2)*a(n-3)))
    end:
seq(a(n), n=0..16);

a(n) = Sum_{j=0..n} (-1)^j*binomial(n,j)*A000680(n-j).
a(n) = A116218(n)/2^n.
a(n) mod 2 = 1 - (n mod 2) = A059841(n).

A116219 If X_1,...,X_n is a partition of a 3n-set X into 3-blocks then a(n) is equal to the number of permutations f of X such that f(X_i) <> X_i, (i=1,...,n).

Original entry on oeis.org

0, 684, 350352, 470444112, 1293433432704, 6355554535465920, 50823027472983319296, 618002474327361540442368, 10855431334634213344062394368, 264600531449039456516679858441216, 8665832467934840277899318819803484160, 371368757645100314808527266212241861300224

Offset: 1

Milan Janjic, Apr 09 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 1..149
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Cf. A116218, A116220, A116221, A127888.

List([1..20], n-> Sum([0..n], j-> (-6)^j*Binomial(n,j)* Factorial(3*n-3*j))); # G. C. Greubel, May 11 2019

[(&+[(-6)^j*Binomial(n, j)*Factorial(3*n-3*j): j in [0..n]]): n in [1..20]]; // G. C. Greubel, May 11 2019

a:=n->sum((-6)^i*binomial(n,i)*(3*n-3*i)!,i=0..n).

Table[Sum[(-6)^i*Binomial[n, i]*(3*n - 3*i)!, {i, 0, n}], {n, 1, 20}] (* G. C. Greubel, May 11 2019 *)

{a(n) = sum(j=0,n, (-6)^j*binomial(n,j)*(3*(n-j))!)}; \\ G. C. Greubel, May 11 2019

[sum((-6)^j*binomial(n,j)*factorial(3*n-3*j) for j in (0..n)) for n in (1..20)] # G. C. Greubel, May 11 2019

a(n) = Sum_{j=0..n} (-6)^j*binomial(n,j)*(3*n-3*j)!.

A116220 If X_1,...,X_n is a partition of a 4n-set X into 4-blocks then a(n) is equal to the number of permutations f of X such that f(X_i) <> X_i, (i=1,...,n).

Original entry on oeis.org

0, 39744, 476126208, 20876944084992, 2430394026897309696, 620098244484731975761920, 304784138698487640049544331264, 263072308376236973471661993731555328, 371936496850567880606221498503260339699712, 815826011669313721421241471652159968573722198016

Offset: 1

Milan Janjic, Apr 09 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 1..110
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Cf. A116218, A116219, A116221.

List([1..20], n-> Sum([0..n], j-> (-24)^j*Binomial(n,j)* Factorial(4*n-4*j))); # G. C. Greubel, May 11 2019

[(&+[(-24)^j*Binomial(n, j)*Factorial(4*n-4*j): j in [0..n]]): n in [1..20]]; // G. C. Greubel, May 11 2019

a:=n->sum((-24)^i*binomial(n,i)*(4*n-4*i)!,i=0..n).

Table[Sum[(-24)^j*Binomial[n, j]*(4*n-4*j)!, {j,0,n}], {n, 1, 20}] (* G. C. Greubel, May 11 2019 *)

{a(n) = sum(j=0,n, (-24)^j*binomial(n,j)*(4*(n-j))!)}; \\ G. C. Greubel, May 11 2019

[sum((-24)^j*binomial(n,j)*factorial(4*n-4*j) for j in (0..n)) for n in (1..20)] # G. C. Greubel, May 11 2019

a(n) = Sum_{j=0..n} (-24)^j*binomial(n,j)*(4*n-4*j)!.

A116221 If X_1,...,X_n is a partition of a 5n-set X into 5-blocks then a(n) is equal to the number of permutations f of X such that f(X_i) <> X_i, (i=1,...,n).

Original entry on oeis.org

0, 3614400, 1306371456000, 2432274637386240000, 15509750490368582860800000, 265241692266421512138485760000000, 10332925158674345473855915900600320000000, 815905363532798455769292988741440720076800000000, 119621339682330952236606797649198078512534126592000000000

Offset: 1

Milan Janjic, Apr 09 2007

nonn

G. C. Greubel, Table of n, a(n) for n = 1..89
Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets

Cf. A116218, A116219, A116220.

List([1..20], n-> Sum([0..n], j-> (-120)^j*Binomial(n,j)* Factorial(5*n-5*j))); # G. C. Greubel, May 11 2019

[(&+[(-120)^j*Binomial(n, j)*Factorial(5*n-5*j): j in [0..n]]): n in [1..20]]; // G. C. Greubel, May 11 2019

a:=n->sum((-120)^i*binomial(n,i)*(5*n-5*i)!,i=0..n).

Table[Sum[(-5!)^j*Binomial[n, j]*(5*n-5*j)!, {j,0,n}], {n,1,20}] (* G. C. Greubel, May 11 2019 *)

{a(n) = sum(j=0,n, (-120)^j*binomial(n,j)*(5*(n-j))!)}; \\ G. C. Greubel, May 11 2019

[sum((-120)^j*binomial(n,j)*factorial(5*n-5*j) for j in (0..n)) for n in (1..20)] # G. C. Greubel, May 11 2019

a(n) = Sum_{j=0..n} (-120)^j*binomial(n,j)*(5*n-5*k)!.

A177840 Consider the n pairs (1,2), ..., (2n-1,2n); a(n) is the number of permutations of [ 2n ] with no two fixed points for any pair.

Original entry on oeis.org

1, 1, 21, 653, 37577, 3434169, 457819549, 83900098309, 20238575173137, 6217167231292913, 2369809434953636261, 1097587512530348834301, 607119566298408076479961, 395312612701784187384578473, 299298318246814086742418737197, 260721599469397754183307347278709

Offset: 0

Paul Weisenhorn, May 14 2010

Inverse binomial transform of (2n)!. - Peter Luschny, May 31 2014

Also, the number of permutations of [2n] with no two cycle (2i-1,2i) for any pair. The number of permutation where no such pair is exchanged or fixed pointwise is A116218. - Aaron Meyerowitz, Jul 22 2023

			a(2) = 21, because there are 4! = 24 permutations of [ 4 ], only 3 of them have pairs with 2 fixed points: [1,2,3,4], [1,2,4,3], [2,1,3,4].
a(3) = A(3,0) = 653, A(3,1) = 63, A(3,2) = 3, A(3,4) = 1, sum = 720 = 6!.

Alois P. Heinz, Table of n, a(n) for n = 0..224
Paul Weisenhorn, First 10 rows of permutation numbers A(n,s) with s=0..n, formatted as a simple linear sequence n, a(n)

Cf. A000166, A064570, A007060, A053983, A053984, A116218.

f:= proc(n) option remember;
      `if`(n<2, 1-n, (n-1) *(f(n-1)+f(n-2)))
    end:
a:= n-> add(binomial(n,j) *2^j *f(2*n-j), j=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Sep 06 2011

f[n_] := f[n] = If[n<2, 1-n, (n-1)*(f[n-1]+f[n-2])]; a[n_] := Sum[Binomial[ n, j]*2^j*f[2*n-j], {j, 0, n}]; Table[a[n], {n, 1, 20}] (* Jean-François Alcover, Feb 25 2017, after Alois P. Heinz *)

a(n) = Sum_{j=0..n} C(n,j) * 2^j * f(2*n-j), where f(n) is the number of permutations of [n] with no fixed-points (A000166).
a(n) = A(n,0), with A(n,s) = number of permutations of [2n] with exactly s pairs with 2 fixed points:
A(n,s) = (n!/s!) * Sum_{j=0..n-s} 1/(j!*(n-s-j)!) * 2^j * f(2*(n-s)-j).
A(n,n) = 1, A(n,n-1) = n, A(n,n-2) = 21*n!/(2*(n-2)!).
Sum_{s=0..n} A(n,s) = (2*n)!.
a(n) = Sum_{j=0..n} C(n,j)*(2*n-2*j)!*(-1)^j. - Tani Akinari, Feb 01 2015
A(n,s) = Sum_{j=s..n} C(n,j)*C(j,s)*(2*n-2*j)!*(-1)^(j-s). - Tani Akinari, Feb 01 2015
From Peter Bala, Mar 07 2015: (Start)
a(n) = Integral_{x = 0..oo} (x^2 - 1)^n*exp(-x) dx.
For n >= 1, Integral_{x = 0..1} (x^2 - 1)^n*exp(x) dx = A007060(n)*e - a(n). Hence lim_{n->oo} a(n)/A007060(n) = e.
O.g.f. with a(0) := 1: Sum_{k >= 0} (2*k)!*x^k/(1 + x)^(k + 1) = 1 + x + 21*x^2 + 653*x^3 + ....
a(n) = 2*n*(2*n - 1)*a(n-1) + 4*n*(n - 1)*a(n-2) + (-1)^n, with initial conditions a(0) = 1, a(1) = 1.
Homogeneous recurrence: a(n) = (4*n^2 - 2*n - 1)*a(n-1) + 2*(n - 1)*(4*n - 3)*a(n-2) + 4*(n - 1)*(n - 2)*a(n-3), with initial conditions a(0) = 1, a(1) = 1 and a(2) = 21. Cf. A064570. (End)
a(n) ~ sqrt(Pi) * 2^(2*n+1) * n^(2*n + 1/2) / exp(2*n). - Vaclav Kotesovec, Mar 10 2015
a(n) = (2*n)!*hypergeom([],[1/2-n],1/4)+(-1)^n*(1-hypergeom([1],[1/2,n+1],1/4)). - Peter Luschny, Mar 15 2015

b-file changed to a-file by N. J. A. Sloane, Oct 05 2010
Edited by Alois P. Heinz, Sep 06 2011
a(0)=1 prepended by Alois P. Heinz, Jul 23 2023

A375220 T(n,k) is the number of permutations of the multiset {1, 1, 2, 2, ..., n, n} with k occurrences of fixed pairs (j,j), where T(n,k), n >= 2, 0 <= k <= n-2 is a triangle read by rows.

Original entry on oeis.org

5, 74, 15, 2193, 296, 30, 101644, 10965, 740, 50, 6840085, 609864, 32895, 1480, 75, 630985830, 47880595, 2134524, 76755, 2590, 105, 76484389121, 5047886640, 191522380, 5692064, 153510, 4144, 140, 11792973495032, 688359502089, 22715489880, 574567140, 12807144, 276318, 6216, 180

Offset: 2

Hugo Pfoertner, Aug 08 2024

			The triangle begins
          5,
         74,       15,
       2193,      296,      30,
     101644,    10965,     740,    50,
    6840085,   609864,   32895,  1480,   75,
  630985830, 47880595, 2134524, 76755, 2590, 105

Cf. A000217, A000680, A028895, A116218, A374980 (column 0), A375222 (column 1), A375223.

Cf. A375219 (similar for triples in the multiset).

\\ using functions mima and a375219 from A375219, row n of triangle:
a375219(n,sizeb=2)

T(n,n) = 1, T(n,n-1) = 0 (terms not in DATA),
T(n,n-2) = 5*n*(n-1)/2 = 5*A000217(n-1) = A028895(n-1),
Sum_{j=0..n-2} T(n,j) = (2*n)!/(2^n) - 1 = A000680(n) - 1,
Sum_{j=1..n-2} T(n,j) = A375223(n) - 1.

cp's OEIS Frontend

A374980 Number of multiset permutations of {1, 1, 2, 2, ..., n, n} with no fixed pair (j,j).

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A116219 If X_1,...,X_n is a partition of a 3n-set X into 3-blocks then a(n) is equal to the number of permutations f of X such that f(X_i) <> X_i, (i=1,...,n).

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A116220 If X_1,...,X_n is a partition of a 4n-set X into 4-blocks then a(n) is equal to the number of permutations f of X such that f(X_i) <> X_i, (i=1,...,n).

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Magma

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A116221 If X_1,...,X_n is a partition of a 5n-set X into 5-blocks then a(n) is equal to the number of permutations f of X such that f(X_i) <> X_i, (i=1,...,n).

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GAP

Magma

Maple

Mathematica

PARI

Sage

Formula

A177840 Consider the n pairs (1,2), ..., (2n-1,2n); a(n) is the number of permutations of [ 2n ] with no two fixed points for any pair.

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A375220 T(n,k) is the number of permutations of the multiset {1, 1, 2, 2, ..., n, n} with k occurrences of fixed pairs (j,j), where T(n,k), n >= 2, 0 <= k <= n-2 is a triangle read by rows.

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