A007838 -id:A007838 - cp's OEIS frontend

A364283 Number of permutations of [n] with distinct cycle lengths such that each cycle contains exactly one cycle length different from its own as an element.

1, 0, 0, 1, 2, 12, 60, 408, 2640, 24480, 208080, 2262960, 23950080, 307359360, 3835641600, 57400358400, 825160089600, 13909727462400, 229664981145600, 4310966499840000, 79428141112320000, 1658163790483200000, 33795850208440320000, 770528520983789568000

Offset: 0

Alois P. Heinz, Jul 17 2023

nonn

			a(3) = 1: (13)(2).
a(4) = 2: (124)(3), (142)(3).
a(5) = 12: (1235)(4), (1253)(4), (1325)(4), (1352)(4), (1523)(4), (1532)(4),
   (124)(35), (142)(35), (125)(34), (152)(34), (13)(245), (13)(254).

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

Cf. A000009, A000166, A007838, A362362, A364277, A364279, A364281, A364282, A364406.

f:= proc(n) option remember; `if`(n<2, 1-n, (n-1)*(f(n-1)+f(n-2))) end:
a:= proc(m) option remember; local b; b:=
      proc(n, i, p) option remember; `if`(i*(i+1)/2

A382781 Sum of GCD of cycle lengths over all permutations of [n] with distinct cycle lengths.

Original entry on oeis.org

0, 1, 2, 9, 32, 170, 1164, 7434, 62880, 582336, 5875200, 60041520, 841501440, 9440926560, 141618778560, 2222190784800, 34862691548160, 543348318159360, 11173101312844800, 186494289764106240, 4219768887634944000, 86094733814301542400, 1834643656963469721600

Offset: 0

Alois P. Heinz, May 11 2025

nonn

			a(3) = 9 = 3+3+1+1+1: (123), (132), (1)(23), (13)(2), (12)(3).

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

Cf. A000142, A007838, A346066, A382780.

b:= proc(n, i, m) option remember; `if`(i*(i+1)/2 b(n$2, 0):
seq(a(n), n=0..22);

A212789 Number of endofunctions on [n] with distinct cycle lengths.

Original entry on oeis.org

1, 1, 3, 20, 186, 2229, 32790, 572018, 11541600, 264370473, 6776462320, 192163455384, 5972728750560, 201906797867085, 7375152706023648, 289473254317393110, 12149690892777901568, 543010240381452000273, 25746662043469525754880, 1290829803802550504743036

Offset: 0

Geoffrey Critzer, May 27 2012

nonn

			a(3)=20 because there are 27 functions f:{1,2,3}->{1,2,3} but 7 of these have at least two cycles of equal length: (1,2,3);(1,2,1);(1,2,2);(1,1,3);(1,3,3);(2,2,3)(3,2,3) where the functions are represented by their values.

Alois P. Heinz, Table of n, a(n) for n = 0..300

Cf. A241980.

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add((i-1)!^j*multinomial(n, n-i*j, i$j)/j!*
      b(n-i*j, i-1), j=0..min(1, n/i))))
    end:
a:= n-> add(binomial(n-1, j-1)*n^(n-j)*b(j$2), j=0..n):
seq(a(n), n=0..25);  # Alois P. Heinz, Aug 10 2014

nn = 20; p = Product[1 + t^n/n, {n, 1, nn}]; t = Sum[n^(n - 1) x^n/n!, {n, 1, nn}]; Range[0, nn]! CoefficientList[Series[p, {x, 0, nn}], x]

E.g.f.: A(T(x)) where A(x) is e.g.f. for A007838 and T(x) is e.g.f. for A000169.

Terms a(8)-a(19) recomputed by Alois P. Heinz, Aug 10 2014

A305203 Expansion of e.g.f. Product_{k>=1} (1 + H(k)*x^k), where H(k) is the k-th harmonic number.

Original entry on oeis.org

1, 1, 3, 20, 94, 854, 7638, 77678, 823184, 11711952, 162710640, 2405290392, 40661618688, 701353671264, 13592382983424, 280431464804640, 5835146351362560, 130171240155651840, 3168997587241864704, 77082927941097660672, 2037627154674197591040, 56017463733173686947840

Offset: 0

Ilya Gutkovskiy, May 27 2018

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..441

Cf. A001008, A002805, A007838, A304494, A305201.

H:= proc(n) H(n):= 1/n +`if`(n=1, 0, H(n-1)) end:
b:= proc(n, i) option remember; `if`(i*(i+1)/2 b(n$2)*n!:
seq(a(n), n=0..25);  # Alois P. Heinz, May 27 2018

nmax = 21; CoefficientList[Series[Product[(1 + HarmonicNumber[k] x^k), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 21; CoefficientList[Series[Exp[Sum[Sum[(-1)^(k + 1) HarmonicNumber[j]^k x^(j k)/k, {j, 1, nmax}], {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, Sum[Sum[(-1)^(k/d + 1) d HarmonicNumber[d]^(k/d), {d, Divisors[k]}] a[n - k], {k, 1, n}]/n]; Table[n! a[n], {n, 0, 21}]

E.g.f.: Product_{k>=1} (1 + (A001008(k)/A002805(k))*x^k).

E.g.f.: exp(Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*H(j)^k*x^(j*k)/k).

A319113 Expansion of e.g.f. Product_{k>=1} (1 + x^prime(k)/prime(k)).

Original entry on oeis.org

1, 0, 1, 2, 0, 44, 0, 1224, 2688, 25920, 293760, 3628800, 25090560, 762048000, 3887170560, 62749209600, 1233908121600, 22616539545600, 321930878976000, 10717413809356800, 108951843667968000, 1982497256570880000, 50138292140310528000, 1408088823809310720000, 25175914255793258496000

Offset: 0

Ilya Gutkovskiy, Sep 10 2018

nonn

Cf. A002099, A007838, A319112.

seq(n!*coeff(series(mul((1+x^ithprime(k)/ithprime(k)),k=1..100),x=0,25),x,n),n=0..24); # Paolo P. Lava, Jan 09 2019

nmax = 24; CoefficientList[Series[Product[(1 + x^Prime[k]/Prime[k]), {k, 1, nmax}], {x, 0, nmax}], x] Range[0, nmax]!
nmax = 24; CoefficientList[Series[Exp[Sum[Sum[Boole[PrimeQ[d]] (-d)^(1 - k/d), {d, Divisors[k]}] x^k/k, {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]!
a[n_] := a[n] = If[n == 0, 1, (n - 1)! Sum[Sum[Boole[PrimeQ[d]] (-d)^(1 - k/d), {d, Divisors[k]}] a[n - k]/(n - k)!, {k, 1, n}]]; Table[a[n], {n, 0, 24}]

my(N=40, x='x+O('x^N)); Vec(serlaplace(prod(k=1, N, 1+isprime(k)*x^k/k))) \\ Seiichi Manyama, Feb 27 2022

E.g.f.: exp(Sum_{k>=1} ( Sum_{p|k, p prime} (-p)^(1-k/p) ) * x^k/k).

A319177 a(n) = n! * [x^n] Product_{k>=1} (1 + x^k/k)^n.

Original entry on oeis.org

1, 1, 4, 39, 500, 7990, 156684, 3640392, 97543088, 2960758800, 100428661440, 3764849536800, 154567280328768, 6897265807262064, 332386213584653760, 17204016957686536320, 951852354201532742400, 56059949872552858763520, 3501729575599545174352896, 231227806715994322631352960

Offset: 0

Ilya Gutkovskiy, Sep 12 2018

nonn

Cf. A007838, A181541, A300187, A319175, A319176.

Table[n! SeriesCoefficient[Product[(1 + x^k/k)^n, {k, 1, n}], {x, 0, n}], {n, 0, 19}]
Table[n! SeriesCoefficient[Exp[n Sum[Sum[(-1)^(k + 1) x^(j k)/(k j^k), {j, 1, n}], {k, 1, n}]], {x, 0, n}], {n, 0, 19}]

a(n) = n! * [x^n] exp(n*Sum_{k>=1} Sum_{j>=1} (-1)^(k+1)*x^(j*k)/(k*j^k)).

A351901 Number of permutations of [n] having at least one repeated cycle length.

Original entry on oeis.org

0, 0, 1, 1, 10, 46, 246, 1926, 16080, 143424, 1397520, 16163280, 190902240, 2534113440, 35501044320, 531674569440, 8558324490240, 147103748144640, 2631981703680000, 50393537347829760, 1011054905709004800, 21229069614652569600, 468171587690550374400

Offset: 0

Alois P. Heinz, Feb 24 2022

nonn

			a(2) = 1: (1)(2).
a(3) = 1: (1)(2)(3).
a(4) = 10: (1)(2)(3)(4), (1)(2)(3,4), (1)(2,4)(3), (1)(2,3)(4), (1,4)(2)(3), (1,3)(2)(4), (1,2)(3)(4), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3).

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

Cf. A000142, A001620, A007838, A227242.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+b(n-i, min(i-1, n-i))/i))
    end:
a:= n-> n!*(1-b(n$2)):
seq(a(n), n=0..23);

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0,
     b[n, i - 1] + b[n - i, Min[i - 1, n - i]]/i]];
a[n_] := n!*(1 - b[n, n]);
Table[a[n], {n, 0, 23}] (* Jean-François Alcover, Apr 19 2022, after Alois P. Heinz *)

E.g.f.: 1/(1-x) - Product_{j>=1} (1 + x^j/j).
a(n) = A000142(n) - A007838(n).
Limit_{n-> infinity} a(n)/n! = 1 - exp(-gamma) = A227242 = 0.43854... .

A382780 Sum of the orders of all permutations of [n] with distinct cycle lengths.

Original entry on oeis.org

1, 1, 2, 12, 48, 360, 2520, 22680, 221760, 2298240, 28425600, 385862400, 5269017600, 80951270400, 1347631084800, 21565729785600, 413922526617600, 8409043612569600, 172028224598630400, 3765253760710041600, 84080417596471296000, 1935910813364656128000

Offset: 0

Alois P. Heinz, May 11 2025

nonn

			a(3) = 12 = 2+2+2+3+3: (1)(23), (13)(2), (12)(3), (123), (132).

Alois P. Heinz, Table of n, a(n) for n = 0..170
Wikipedia, Permutation

Cf. A000142, A000793, A007838, A060014, A382781.

b:= proc(n, i, m) option remember; `if`(i*(i+1)/2 b(n$2, 1):
seq(a(n), n=0..22);

A131622 Number of cycles in all permutations of n elements with distinct cycle lengths.

Original entry on oeis.org

1, 1, 8, 22, 124, 948, 6138, 50832, 468144, 5165280, 54704880, 695854080, 9016051680, 130427750880, 1994479744320, 32575206343680, 555499414471680, 10284817657927680, 196642556903116800, 3994718386866278400, 84989047758544742400, 1895851170953432985600

Offset: 1

Vladeta Jovovic, Sep 02 2007

Alois P. Heinz, Table of n, a(n) for n = 1..450

Cf. A000254, A007838.

A131622 := proc(n) local su,i ; su := add(x^i/(i+x^i),i=1..n+1) ; for i from 1 to n do su := taylor(su*(1+x^i/i),x=0,n+1) ; od: n!*coeftayl(su,x=0,n) ; end: seq(A131622(n),n=1..30) ; # R. J. Mathar, Oct 25 2007
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0],
      `if`(i<1, 0, `if`(i>n, 0, (p->[0, p[1]]+p)(
       b(n-i, i-1)*binomial(n, i)*(i-1)!))+b(n, i-1)))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=1..30);  # Alois P. Heinz, May 14 2016

b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i < 1, {0, 0}, If[i > n, {0, 0}, Function[p, {0, p[[1]]} + p][b[n-i, i-1] Binomial[n, i] (i-1)!]] + b[n, i-1]]];
a[n_] := b[n, n][[2]];
Array[a, 30] (* Jean-François Alcover, May 22 2020, after Alois P. Heinz *)
nmax = 30; Rest[CoefficientList[Series[Sum[x^k/(k + x^k), {k, 1, nmax}] * Product[1 + x^k/k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]!] (* Vaclav Kotesovec, May 22 2020 *)

E.g.f.: Sum(x^n/(n+x^n), n=1..inf) * Product(1+x^n/n, n=1..inf).

More terms from R. J. Mathar, Oct 25 2007

A224211 Irregular triangular array read by rows. T(n,k) is the number of n-permutations with exactly k distinct cycle lengths; n>=1, 1<=k<=floor( (-1+(1+8n)^(1/2))/2 ).

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 24, 50, 120, 234, 120, 720, 1764, 630, 5040, 11808, 7392, 40320, 109584, 69552, 362880, 954000, 763200, 151200, 3628800, 10628640, 8165520, 1330560, 39916800, 113891040, 109010880, 25280640, 479001600, 1486442880, 1345687200, 381775680, 6227020800, 18913184640, 19773804960, 6763236480, 87178291200, 283465647360, 291950568000, 102508005600, 10897286400

Offset: 1

Geoffrey Critzer, Apr 01 2013

Row sums = A007838.

			:      1;
:      1;
:      2,      3;
:      6,      8;
:     24,     50;
:    120,    234,    120;
:    720,   1764,    630;
:   5040,  11808,   7392;
:  40320, 109584,  69552;
: 362880, 954000, 763200, 151200;

Alois P. Heinz, Rows n = 1..300, flattened

b:= proc(n, i) option remember; expand(`if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+ (i-1)!*b(n-i, i-1)*
      `if`(i>n, 0, binomial(n, i)*x))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n$2)):
seq(T(n), n=1..15);  # Alois P. Heinz, Oct 21 2015

nn=15;f[list_]:=Select[list,#>0&];Map[f,Drop[Range[0,nn]!CoefficientList[Series[Product[(1+y x^i/i),{i,1,nn}],{x,0,nn}],{x,y}],1]]//Grid

E.g.f.: Product_{i>=1} (1 + y*x)^i/i.

cp's OEIS Frontend

A364283 Number of permutations of [n] with distinct cycle lengths such that each cycle contains exactly one cycle length different from its own as an element.

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A382781 Sum of GCD of cycle lengths over all permutations of [n] with distinct cycle lengths.

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Maple

A212789 Number of endofunctions on [n] with distinct cycle lengths.

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Extensions

A305203 Expansion of e.g.f. Product_{k>=1} (1 + H(k)*x^k), where H(k) is the k-th harmonic number.

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A319113 Expansion of e.g.f. Product_{k>=1} (1 + x^prime(k)/prime(k)).

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A319177 a(n) = n! * [x^n] Product_{k>=1} (1 + x^k/k)^n.

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Mathematica

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A351901 Number of permutations of [n] having at least one repeated cycle length.

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Maple

Mathematica

Formula

A382780 Sum of the orders of all permutations of [n] with distinct cycle lengths.

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Maple

A131622 Number of cycles in all permutations of n elements with distinct cycle lengths.