Luis Manuel Rivera Martínez - cp's OEIS frontend

User: Luis Manuel Rivera Martínez

Luis Manuel Rivera Martínez's wiki page.

Luis Manuel Rivera Martínez has authored 4 sequences.

A326241 Number of degree-n even permutations of order dividing 12.

1, 1, 1, 3, 12, 36, 216, 1296, 10368, 78912, 634896, 5572656, 51817536, 477672768, 8268884352, 101752505856, 1417554660096, 20985416983296, 344834432195328, 5096129755468032, 70148917686998016

Offset: 0

Luis Manuel Rivera Martínez, Jul 06 2019

			For n=3 the a(3)=3 solutions are (1), (1, 2, 3), (1, 3, 2) (permutations in cyclic notation).

J. Riordan, An Introduction to Combinatorial Analysis, John Wiley & Sons, Inc. New York, 1958 (Chap 4, Problem 22).

Robert Israel, Table of n, a(n) for n = 0..482
Lev Glebsky, Melany Licón, Luis Manuel Rivera, On the number of even roots of permutations, arXiv:1907.00548 [math.CO], 2019.

Cf. A053502, A326242, A000704, A061130, A061131, A061132, A048099, A051695, A061133, A061134, A061135, A326242.

E:= (1/2)*exp(x + (1/2)*x^2 + (1/3)*x^3 + (1/4)*x^4 + (1/6)*x^6+(1/12)*x^(12)) + (1/2)*exp(x - (1/2)*x^2 + (1/3)*x^3 - (1/4)*x^4 - (1/6)*x^6-(1/12)*x^(12)):
S:= series(E,x,31):
seq(coeff(S,x,i)*i!,i=0..30);# Robert Israel, Jul 08 2019

With[{nn = 22}, CoefficientList[Series[1/2 Exp[x + x^2/2 + x^3/3 + x^4/4 + x^6/6 +x^12/12]+1/2 Exp[x - x^2/2 + x^3/3 - x^4/4 - x^6/6 - x^12/12], {x, 0, nn}], x]*Range[0, nn]!]

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

A326242 Number of degree-n odd permutations of order dividing 12.

Original entry on oeis.org

0, 0, 1, 3, 12, 60, 360, 2016, 11088, 73872, 602640, 4411440, 81677376, 934435008, 8100473472, 104370819840, 1448725616640, 15823660179456, 247231858514688, 3703908371910912, 66727356304757760, 1124506454958351360, 19305439846610835456

Offset: 0

Luis Manuel Rivera Martínez, Jul 06 2019

nonn

			For n=3 the a(3)=3 solutions are (1, 2), (2, 3), (1, 3) (permutations in cyclic notation).

Robert Israel, Table of n, a(n) for n = 0..482
Lev Glebsky, Melany Licón, Luis Manuel Rivera, On the number of even roots of permutations, arXiv:1907.00548 [math.CO], 2019.

Cf. A053502, A326242, A000704, A061130, A061131, A061132, A048099, A051695, A061133, A061134, A061135, A326241.

E:= (1/2)*exp(x + (1/2)*x^2 + (1/3)*x^3 + (1/4)*x^4 + (1/6)*x^6+(1/12)*x^(12)) - (1/2)*exp(x - (1/2)*x^2 + (1/3)*x^3 - (1/4)*x^4 - (1/6)*x^6-(1/12)*x^(12)):
S:= series(E,x,31):
seq(coeff(S,x,i)*i!,i=0..30); # Robert Israel, Jul 08 2019

With[{nn = 22}, CoefficientList[Series[1/2 Exp[x + x^2/2 + x^3/3 + x^4/4 + x^6/6 +x^12/12]-1/2 Exp[x - x^2/2 + x^3/3 - x^4/4 - x^6/6 - x^12/12], {x, 0, nn}], x]*Range[0, nn]!]

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

A308648 Number of degree-n odd permutations of order dividing 8.

Original entry on oeis.org

0, 0, 1, 3, 12, 40, 120, 336, 7168, 58752, 345600, 1682560, 15983616, 142192128, 2318697472, 25614382080, 282753361920, 2645093410816, 48869743454208, 674729909839872, 12153962014842880, 167314499427532800, 1986101341059956736, 20335611320801886208

Offset: 0

Luis Manuel Rivera Martínez, Jun 14 2019

			For n=3 the a(3)=3 solutions are (1, 2), (1, 3), (2, 3) (permutations in cyclic notation).

Cf. A061131, A061136, A061137.

With[{nn = 22},
CoefficientList[Series[1/2 Exp[x + x^2/2 + x^4/4 + x^8/8] - 1/2 Exp[x - x^2/2 - x^4/4 - x^8/8], {x, 0, nn}], x]*Range[0, nn]!]

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

A233440 Triangle read by rows: T(n, k) = nbinomial(n, k)A000757(k), 0 <= k <= n.

Original entry on oeis.org

0, 1, 0, 2, 0, 0, 3, 0, 0, 3, 4, 0, 0, 16, 4, 5, 0, 0, 50, 25, 40, 6, 0, 0, 120, 90, 288, 216, 7, 0, 0, 245, 245, 1176, 1764, 1603, 8, 0, 0, 448, 560, 3584, 8064, 14656, 13000, 9, 0, 0, 756, 1134, 9072, 27216, 74196, 131625, 118872, 10, 0, 0, 1200, 2100, 20160, 75600, 274800, 731250, 1320800, 1202880

Offset: 0

Luis Manuel Rivera Martínez, Dec 09 2013

For n >= 0, 0 <= k <= n, T(n, k) is the number of permutations of n symbols that k-commute with an n-cycle (we say that two permutations f and g k-commute if H(fg, gf) = k, where H(, ) denotes the Hamming distance between permutations).

Row sums give A000142.

			For n = 4 and k = 4, T(4, 4) = 4 because all the permutations of 4 symbols that 4-commute with permutation (1, 2, 3, 4) are (1, 3), (2, 4), (1, 2)(3, 4) and (1, 4)(2, 3).

Luis Manuel Rivera Martínez, Rows n = 0..30 of triangle, flattened
R. Moreno and L. M. Rivera, Blocks in cycles and k-commuting permutations, arXiv:1306.5708 [math.CO], 2013-2014.
Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.

Cf. A007318, A000757.

T[n_, k_] := n Binomial[n, k] ((-1)^k+Sum[(-1)^j k!/(k-j)/j!, {j, 0, k-1}]);
Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Nov 03 2018 *)

T(n,k) = n*C(n,k)*A000757(k), 0 <= k <= n.
Bivariate e.g.f.: G(z, u) = z*exp(z*(1-u))*(u/(1-z*u)+(1-log(1-z*u))*(1-u)).
T(n, 0)  = A001477(n), n>=0;
T(n, 1)  = A000004(n), n>=1;
T(n, 2)  = A000004(n), n>=2;
T(n, 3)  = A004320(n-2), n>=3;
T(n, 4)  = A027764(n-1), n>=4;
T(n, 5)  = A027765(n-1)*A000757(5), n>=5;
T(n, 6)  = A027766(n-1)*A000757(6), n>=6;
T(n, 7)  = A027767(n-1)*A000757(7), n>=7;
T(n, 8)  = A027768(n-1)*A000757(8), n>=8;
T(n, 9)  = A027769(n-1)*A000757(9), n>=9;
T(n, 10) = A027770(n-1)*A000757(10), n>=10;
T(n, 11) = A027771(n-1)*A000757(11), n>=11;
T(n, 12) = A027772(n-1)*A000757(12), n>=12;
T(n, 13) = A027773(n-1)*A000757(13), n>=13;
T(n, 14) = A027774(n-1)*A000757(14), n>=14;
T(n, 15) = A027775(n-1)*A000757(15), n>=15;
T(n, 16) = A027776(n-1)*A000757(16), n>=16. - Luis Manuel Rivera Martínez, Feb 08 2014
T(n, 0)+T(n, 3) = n*A050407(n+1), for n>=0. - Luis Manuel Rivera Martínez, Mar 06 2014

cp's OEIS Frontend

User: Luis Manuel Rivera Martínez

A326241 Number of degree-n even permutations of order dividing 12.

Views

Author

Keywords

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A326242 Number of degree-n odd permutations of order dividing 12.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A308648 Number of degree-n odd permutations of order dividing 8.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula

A233440 Triangle read by rows: T(n, k) = n*binomial(n, k)*A000757(k), 0 <= k <= n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A233440 Triangle read by rows: T(n, k) = nbinomial(n, k)A000757(k), 0 <= k <= n.