A003483 -id:A003483 - cp's OEIS frontend

A214849 Number of n-permutations having all cycles of odd length and at most one fixed point.

1, 1, 0, 2, 8, 24, 184, 1000, 8448, 66752, 670976, 6771456, 80540800, 981684352, 13555365888, 193136762624, 3042586824704, 49558509465600, 877951349198848, 16081833643651072, 316609129672114176, 6439690754082062336, 139521103623589068800

Offset: 0

Geoffrey Critzer, Mar 08 2013

nonn

a(n) is also the number of n-permutations with exactly one square root. Cf. A003483 which counts n-permutations with at least one square root.

			a(6)= 184 because we have 144 6-permutations of the type (1,2,3,4,5)(6) and 40 of the type (1,2,3)(4,5,6).  These have exactly one square root: (1,4,2,5,3)(6) and (1,3,2)(4,6,5).

Vincenzo Librandi, Table of n, a(n) for n = 0..200

nn=22; Range[0,nn]! CoefficientList[Series[(1+x)((1+x)/(1-x))^(1/2) Exp[-x], {x,0,nn}], x]

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

a(n) ~ 4*n^n/exp(n+1). - Vaclav Kotesovec, Oct 08 2013

A102760 Number of partitions of n-set into "lists", in which every even list appears an even number of times, cf. A000262.

Original entry on oeis.org

1, 1, 1, 7, 37, 241, 1381, 13231, 140617, 1483777, 16211881, 217551511, 3384215341, 50221272817, 782154787597, 13913712591871, 272739557719441, 5282625708305281, 106588332600443857, 2354480141600267047, 56238135934525073461, 1338131691952924913521

Offset: 0

Vladeta Jovovic, Feb 10 2005

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..445

Cf. A003483, A006950, A052565, A088026.

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(`if`(i::even and j::odd, 0, b(n-i*j, i-1)*
      multinomial(n, n-i*j, i$j)/j!*i!^j), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..25);  # Alois P. Heinz, May 10 2016

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Sum[If[EvenQ[i] && OddQ[j], 0, b[n-i*j, i- 1] * multinomial[n, Join[{n - i*j}, Array[i &, j]]]/j!*i!^j], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 05 2017, after Alois P. Heinz *)

E.g.f.: exp(x/(1-x^2))*Product_{k>0} cosh(x^(2*k)).

a(0)=1 prepended by Alois P. Heinz, May 10 2016

A214851 Irregular triangular array read by rows. T(n,k) is the number of n-permutations that have exactly k square roots. n >= 1, 0 <= k <= A000085(n).

Original entry on oeis.org

0, 1, 1, 0, 1, 3, 2, 0, 0, 1, 12, 8, 3, 0, 0, 0, 0, 0, 0, 0, 1, 60, 24, 35, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 450, 184, 0, 0, 85, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Geoffrey Critzer, Mar 08 2013

Row sums = n!.
Sum_{k=1...A000085(n)} T(n,k)*k = n!.
Sum_{k=1...A000085(n)} T(n,k) = A003483(n).
Column k=0 is n! - A003483(n).

			0, 1,
1, 0, 1,
3, 2, 0, 0, 1,
12, 8, 3, 0, 0, 0, 0, 0, 0, 0, 1,
60, 24, 35, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1
450, 184, 0, 0, 85, 0,0,0,...,1 where the 1 is in column k=76.
T(5,2)= 35 because we have 20 5-permutations of the type (1,2,3)(4)(5) and 15 of the type (1,2)(3,4)(5).  These have 2 square roots:(1,3,2)(4)(5),(1,3,2)(4,5) and (1,3,2,4)(5),(3,1,4,2)(5) respectively.

Cf. A214849 (column k=1), A214854 (column k=2).

(* Warning: the code is very inefficient, it takes about one minute to run on a laptop computer. *) a={1,2,4,10,26}; Table[Distribution[Distribution[Table[MultiplicationTable[Permutations[m], Permute[#1,#2]&][[n]][[n]], {n,1,m!}], Range[1,m!]], Range[0,a[[m]]]], {m,1,5}] //Grid

A130275 Number of degree-n permutations such that number of cycles of size 2k is odd (or zero) for every k.

Original entry on oeis.org

1, 1, 2, 6, 21, 105, 675, 4725, 35805, 322245, 3236625, 35602875, 425872755, 5536345815, 77347084815, 1160206272225, 18403556596425, 312860462139225, 5643104418376425, 107218983949152075, 2136610763952639975, 44868826043005439475, 986129980012277775675

Offset: 0

Vladeta Jovovic, Aug 06 2007

			a(4)=21 because only the following three degree-4 permutations do not qualify: (12)(34), (13)(24) and (14)(23).

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A003483, A006950, A015128, A102759, A130126, A131942, A130219-A130223.

g:=sqrt((1+x)/(1-x))*(product(1+sinh(x^(2*k)/(2*k)),k=1..30)): gser:=series(g, x=0,25): seq(factorial(n)*coeff(gser,x,n),n=0..20); # Emeric Deutsch, Aug 24 2007
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      `if`(j=0 or irem(i, 2)=1 or irem(j, 2)=1, multinomial(n,
       n-i*j, i$j)*(i-1)!^j/j!*b(n-i*j, i-1), 0), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 09 2015

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[If[j == 0 || Mod[i, 2] == 1 || Mod[j, 2] == 1, multinomial[n, Join[{n - i*j}, Array[i &, j]]]*(i - 1)!^j/j!*b[n - i*j, i - 1], 0], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Dec 22 2016, after Alois P. Heinz *)

E.g.f.: sqrt((1+x)/(1-x))*Product_{k>0} (1+sinh(x^(2*k)/(2*k))).

More terms from Emeric Deutsch, Aug 24 2007

A130276 Number of degree-2n permutations such that number of cycles of size 2k-1 is even (or zero) for every k.

Original entry on oeis.org

1, 2, 16, 416, 20224, 1645312, 196388864, 33279311872, 7427338829824, 2151276556845056, 771086221948223488, 340572557390992900096, 179222835344084459061248, 112158801651454395931426816, 81399358513573250066141937664, 68530340884909785149816189222912

Offset: 0

Vladeta Jovovic, Aug 06 2007

			a(2)=16 because there are 8 permutations that do not qualify: (1)(234), (1)(243), (123)(4), (124)(3), (132)(4), (134)(2), (142)(3) and (143)(2).

Alois P. Heinz, Table of n, a(n) for n = 0..220

Cf. A003483, A006950, A015128, A102759, A130126, A131942, A130219-A130223.

g:=(product(cosh(x^(2*k-1)/(2*k-1)),k=1..30))/sqrt(1-x^2): gser:=series(g,x= 0,30): seq(factorial(2*n)*coeff(gser,x,2*n),n=0..13); # Emeric Deutsch, Aug 24 2007
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      `if`(j=0 or irem(i, 2)=0 or irem(j, 2)=0, multinomial(n,
       n-i*j, i$j)*(i-1)!^j/j!*b(n-i*j, i-1), 0), j=0..n/i)))
    end:
a:= n-> b(2*n$2):
seq(a(n), n=0..20);  # Alois P. Heinz, Mar 09 2015

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[If[j == 0 || Mod[i, 2] == 0 || Mod[j, 2] == 0, multinomial[n, Join[{n - i*j}, Array[i &, j]]]*(i - 1)!^j/j!*b[n - i*j, i - 1], 0], {j, 0, n/i}]]]; a[n_] := b[2n, 2n]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Dec 22 2016, after Alois P. Heinz *)

N=31; x='x+O('x^N);
v0=Vec(serlaplace(1/sqrt(1-x^2)*prod(k=1,N, cosh(x^(2*k-1)/(2*k-1)))));
vector(#v0\2,n,v0[2*n-1]) \\ Joerg Arndt, Jan 03 2011

E.g.f. with interleaved zeros: 1/sqrt(1-x^2)*Product_{k>=1} cosh(x^(2*k-1)/(2*k-1)). - Geoffrey Critzer, Jan 02 2011

More terms from Emeric Deutsch, Aug 24 2007

A130278 Number of degree-n permutations such that number of cycles of size 2k-1 is odd (or zero) for every k.

Original entry on oeis.org

1, 1, 1, 6, 17, 100, 529, 3766, 31121, 276984, 2755553, 29665306, 364627801, 4639937380, 64952094401, 973467571350, 15750475301921, 264870218828656, 4759194994114369, 90124395399063730, 1812001488739061417, 37956199941196210716, 832297726351555617569

Offset: 0

Vladeta Jovovic, Aug 06 2007

			a(4)=17 because only the following 7 permutations do not qualify: (1)(2)(3)(4), (1)(2)(34), (1)(23)(4), (1)(24)(3), (12)(3)(4), (13)(2)(4) and (14)(2)(3).

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A003483, A006950, A015128, A102759, A130126, A131942, A130219-A130223.

g:=(product(1+sinh(x^(2*k-1)/(2*k-1)),k=1..30))/sqrt(1-x^2): gser:=series(g,x =0,25): seq(factorial(n)*coeff(gser,x,n),n=0..20); # Emeric Deutsch, Aug 24 2007
# second Maple program:
with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0, add(
      `if`(j=0 or irem(i, 2)=0 or irem(j, 2)=1, multinomial(n,
       n-i*j, i$j)*(i-1)!^j/j!*b(n-i*j, i-1), 0), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 09 2015

multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[If[j == 0 || Mod[i, 2] == 0 || Mod[j, 2] == 1, multinomial[n, Join[{n - i*j}, Array[i&, j]]]*(i - 1)!^j/j!*b[n - i*j, i - 1], 0], {j, 0, n/i}]]]; a[n_] := b[n, n]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Dec 22 2016, after Alois P. Heinz *)

E.g.f.: 1/sqrt(1-x^2)*Product_{k>0} (1+sinh(x^(2*k-1)/(2*k-1))).

More terms from Emeric Deutsch, Aug 24 2007

A155510 Possible cardinalities of the set of all k-th powers of the order n permutations, where k and n are some positive integers.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 12, 16, 21, 24, 25, 36, 40, 45, 46, 56, 60, 80, 81, 96, 106, 120, 126, 145, 190, 225, 256, 270, 351, 400, 505, 576, 610, 666, 720, 721, 826, 855, 946, 1071, 1072, 1170, 1225, 1233, 1330, 1338, 1345, 1386, 1450, 1575, 1576, 1792, 1890, 2080, 2241

Offset: 1

Vladimir Letsko, Jan 23 2009

nonn

			80 is in the sequence because the set {a^3|a in S_5} has 80 elements.

Vladimir Letsko, Mathematical Marathon at VSPU, Problem 100 (in Russian)
Vladimir Letsko, Mathematical Marathon at dxdy (in Russian)

Set of elements of A247005.

Cf. A000012, A000142, A003483, A103619, A103620, A215716, A215717, A215718, A247006, A247007, A247008, A247009.

Corrected and extended by Max Alekseyev, Feb 08 2009

Some missing terms added by Max Alekseyev, Jan 24 2010

A214854 Number of n-permutations that have exactly two square roots.

Original entry on oeis.org

0, 0, 1, 0, 3, 35, 0, 714, 2835, 35307, 236880, 3342350, 28879158, 461911086, 4916519608, 87798024300, 1112716544355, 21957112744083, 322944848419392, 6986165252185782, 116941654550250258, 2754405555107729418, 51688464405692879688

Offset: 0

Geoffrey Critzer, Mar 08 2013

nonn

These permutations are of two types: They are composed of exactly one pair of equal even size cycles with at most one fixed point and any number of odd (>=3) size cycles; OR they are any number of odd (>=3) size cycles with exactly two fixed points.

			a(5) = 35 because we have 20 5-permutations of the type (1,2,3)(4)(5) and 15 of the type (1,2)(3,4)(5). These have 2 square roots:(1,3,2)(4)(5),(1,3,2)(4,5) and (1,3,2,4)(5),(3,1,4,2)(5) respectively.

Cf. A214849, A214851, A003483.

nn=22; a=Sum[Binomial[2n,n]/2x^(2n)/(2n)!, {n,2,nn,2}]; Range[0,nn]! CoefficientList[Series[(a(1+x)+x^2/2) ((1+x)/(1-x))^(1/2) Exp[-x], {x,0,nn}], x]

E.g.f.: (A(x)*(1+x)+x^2/2)*((1+x)/(1-x))^(1/2)*exp(-x) where A(x) = Sum_{n=2,4,6,8,...} Binomial(2n,n)/2 * x^(2n)/(2n)!

A247621 Decimal expansion of Sum_{k>=1} 1/(2k)-tanh(1/(2k)).

Original entry on oeis.org

0, 4, 6, 1, 5, 1, 4, 5, 1, 3, 0, 5, 9, 1, 4, 2, 0, 8, 8, 4, 6, 5, 6, 8, 3, 8, 3, 5, 4, 1, 7, 4, 6, 7, 4, 7, 2, 8, 2, 7, 4, 9, 5, 5, 9, 7, 4, 7, 1, 0, 5, 3, 7, 8, 1, 4, 7, 8, 9, 0, 9, 7, 0, 3, 8, 0, 2, 8, 3, 4, 5, 1, 1, 2, 4, 8, 7, 9, 5, 2, 0, 1, 9, 8, 2, 7, 0, 4, 0, 5, 0, 1, 1, 8, 8, 9, 5, 0, 1, 2, 6, 8, 8, 9, 7, 2

Offset: 0

Vaclav Kotesovec, Sep 21 2014

			0.04615145130591420884656838354174674728274955974710537814789097038...

Philippe Flajolet, Éric Fusy, Xavier Gourdon, Daniel Panario and Nicolas Pouyanne, A Hybrid of Darboux's Method and Singularity Analysis in Combinatorial Asymptotics, arXiv:math.CO/0606370, p. 18

Cf. A003483.

evalf(sum(1/(2*k)-tanh(1/(2*k)), k=1..infinity), 120)

default(realprecision,150); sumpos(k=1,(1/(2*k)-tanh(1/(2*k))))

A349645 Triangular array read by rows: T(n,k) is the number of square n-permutations possessing exactly k cycles; n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 11, 0, 1, 0, 24, 0, 35, 0, 1, 0, 0, 184, 0, 85, 0, 1, 0, 720, 0, 994, 0, 175, 0, 1, 0, 0, 9708, 0, 4249, 0, 322, 0, 1, 0, 40320, 0, 72764, 0, 14889, 0, 546, 0, 1, 0, 0, 648576, 0, 402380, 0, 44373, 0, 870, 0, 1

Offset: 0

Steven Finch, Nov 23 2021

A permutation p in S_n is a square if there exists q in S_n with q^2=p.

For such a p, the number of cycles of any even length in its disjoint cycle decomposition must be even.

			The three square 3-permutations are (1, 2, 3) with three cycles (fixed points) and (3, 1, 2) & (2, 3, 1), each with one cycle.
Among the twelve square 4-permutations are {1, 4, 2, 3} & {1, 3, 4, 2} and {3, 4, 1, 2} & {4, 3, 2, 1}, all with two cycles but differing types.
Triangle begins:
[0]   1;
[1]   0,   1;
[2]   0,   0,    1;
[3]   0,   2,    0,   1;
[4]   0,   0,   11,   0,    1;
[5]   0,  24,    0,  35,    0,   1;
[6]   0,   0,  184,   0,   85,   0,   1;
[7]   0, 720,    0, 994,    0, 175,   0,   1;
[8]   0,   0, 9708,   0, 4249,   0, 322,   0,   1;
...

Alois P. Heinz, Rows n = 0..200, flattened
Steven Finch, Rounds, Color, Parity, Squares, arXiv:2111.14487 [math.CO], 2021.

Columns k=0-1 give: A000007, A005359(n-1).
Row sums give A003483.
T(n+2,n) gives A000914.
Cf. A214851, A246945.

with(combinat):
b:= proc(n, i) option remember; expand(`if`(n=0, 1, `if`(i<1, 0,
      add(`if`(irem(i, 2)=0 and irem(j, 2)=1, 0, (i-1)!^j*
      multinomial(n, n-i*j, i$j)/j!*b(n-i*j, i-1))*x^j, j=0..n/i))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n$2)):
seq(T(n), n=0..12);  # Alois P. Heinz, Nov 23 2021

multinomial[n_, k_List] := n!/Times @@ (k!);
b[n_, i_] := b[n, i] = Expand[If[n == 0, 1, If[i < 1, 0,
     Sum[If[Mod[i, 2] == 0 && Mod[j, 2] == 1, 0, (i-1)!^j*multinomial[n,
     Join[{n-i*j}, Table[i, {j}]]]/j!*b[n-i*j, i-1]]*x^j, {j, 0, n/i}]]]];
T[n_] := With[{p = b[n, n]}, Table[Coefficient[p, x, i], {i, 0, n}]];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 28 2021, after Alois P. Heinz *)

cp's OEIS Frontend

A214849 Number of n-permutations having all cycles of odd length and at most one fixed point.

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A102760 Number of partitions of n-set into "lists", in which every even list appears an even number of times, cf. A000262.

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A214851 Irregular triangular array read by rows. T(n,k) is the number of n-permutations that have exactly k square roots. n >= 1, 0 <= k <= A000085(n).

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A130275 Number of degree-n permutations such that number of cycles of size 2k is odd (or zero) for every k.

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Mathematica

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A130276 Number of degree-2n permutations such that number of cycles of size 2k-1 is even (or zero) for every k.

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PARI

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A130278 Number of degree-n permutations such that number of cycles of size 2k-1 is odd (or zero) for every k.

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Maple

Mathematica

Formula

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A155510 Possible cardinalities of the set of all k-th powers of the order n permutations, where k and n are some positive integers.

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Examples

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Extensions

A214854 Number of n-permutations that have exactly two square roots.

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A247621 Decimal expansion of Sum_{k>=1} 1/(2k)-tanh(1/(2k)).