A028418 -id:A028418 - cp's OEIS frontend

A126074 Triangle read by rows: T(n,k) is the number of permutations of n elements that have the longest cycle length k.

1, 1, 1, 1, 3, 2, 1, 9, 8, 6, 1, 25, 40, 30, 24, 1, 75, 200, 180, 144, 120, 1, 231, 980, 1260, 1008, 840, 720, 1, 763, 5152, 8820, 8064, 6720, 5760, 5040, 1, 2619, 28448, 61236, 72576, 60480, 51840, 45360, 40320, 1, 9495, 162080, 461160, 653184, 604800, 518400, 453600, 403200, 362880

Offset: 1

Dan Dima, Mar 01 2007

Sum of the n-th row is the number of all permutations of n elements: Sum_{k=1..n, T(n,k)} = n! = A000142(n) We can extend T(n,k)=0, if k<=0 or k>n.
From Peter Luschny, Mar 07 2009: (Start)
Partition product of prod_{j=0..n-2}(k-n+j+2) and n! at k = -1, summed over parts with equal biggest part (see the Luschny link).
Underlying partition triangle is A102189.
Same partition product with length statistic is A008275.
Diagonal a(A000217(n)) = rising_factorial(1,n-1), A000142(n-1) (n > 0).
Row sum is A000142. (End)
Let k in {1,2,3,...} index the family of sequences A000012, A000085, A057693, A070945, A070946, A070947, ... respectively. Column k is the k-th sequence minus its immediate predecessor. For example, T(5,3)=A057693(5)-A000085(5). - Geoffrey Critzer, May 23 2009

			Triangle T(n,k) begins:
  1;
  1,   1;
  1,   3,    2;
  1,   9,    8,    6;
  1,  25,   40,   30,   24;
  1,  75,  200,  180,  144,  120;
  1, 231,  980, 1260, 1008,  840,  720;
  1, 763, 5152, 8820, 8064, 6720, 5760, 5040;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Steven Finch, Permute, Graph, Map, Derange, arXiv:2111.05720 [math.CO], 2021.
S. W. Golomb and P. Gaal, On the number of permutations of n objects with greatest cycle length k, Adv. in Appl. Math., 20(1), 1998, 98-107.
IBM Research, Ponder This, December 2006.
Peter Luschny, Counting with Partitions. [From _Peter Luschny_, Mar 07 2009]
Peter Luschny, Generalized Stirling_1 Triangles. [From _Peter Luschny_, Mar 07 2009]
D. Panario and B. Richmond, Exact largest and smallest size of components, Algorithmica, 31 (2001), 413-432.

Cf. A000142.
Cf. A071007, A080510, A028418.
Cf. A157386, A157385, A157384, A157383, A157400, A157391, A157392, A157393, A157394, A157395. - Peter Luschny, Mar 07 2009
T(2n,n) gives A052145 (for n>0). - Alois P. Heinz, Apr 21 2017

A:= proc(n,k) option remember; `if`(n<0, 0, `if`(n=0, 1,
       add(mul(n-i, i=1..j-1)*A(n-j,k), j=1..k)))
    end:
T:= (n, k)-> A(n, k) -A(n, k-1):
seq(seq(T(n,k), k=1..n), n=1..10);  # Alois P. Heinz, Feb 11 2013

Table[CoefficientList[ Series[(Exp[x^m/m] - 1) Exp[Sum[x^k/k, {k, 1, m - 1}]], {x, 0, 8}], x]*Table[n!, {n, 0, 8}], {m, 1, 8}] // Transpose // Grid (* Geoffrey Critzer, May 23 2009 *)

def A126074(n, k):
    f = factorial(n)
    P = Partitions(n, max_part=k, inner=[k])
    return sum(f // p.aut() for p in P)
for n in (1..9): print([A126074(n,k) for k in (1..n)]) # Peter Luschny, Apr 17 2016

T(n,1) = 1.
T(n,2) = n! * Sum_{k=1..[n/2]} 1/(k! * (2!)^k * (n-2*k)!).
T(n,k) = n!/k * (1-1/(n-k)-...-1/(k+1)-1/2k), if n/3 < k <= n/2.
T(n,k) = n!/k, if n/2 < k <= n.
T(n,n) = (n-1)! = A000142(n-1).
E.g.f. for k-th column: exp(-x^k*LerchPhi(x,1,k))*(exp(x^k/k)-1)/(1-x). - Vladeta Jovovic, Mar 03 2007
From Peter Luschny, Mar 07 2009: (Start)
T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n
T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that
1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!),
f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-2}(j-n+1). (End)
Sum_{k=1..n} k * T(n,k) = A028418(n). - Alois P. Heinz, May 17 2016

A322384 Number T(n,k) of entries in the k-th cycles of all permutations of [n] when cycles are ordered by decreasing lengths (and increasing smallest elements); triangle T(n,k), n>=1, 1<=k<=n, read by rows.

Original entry on oeis.org

1, 3, 1, 13, 4, 1, 67, 21, 7, 1, 411, 131, 46, 11, 1, 2911, 950, 341, 101, 16, 1, 23563, 7694, 2871, 932, 197, 22, 1, 213543, 70343, 26797, 9185, 2311, 351, 29, 1, 2149927, 709015, 275353, 98317, 27568, 5119, 583, 37, 1, 23759791, 7867174, 3090544, 1141614, 343909, 73639, 10366, 916, 46, 1

Offset: 1

Alois P. Heinz, Dec 05 2018

			The 6 permutations of {1,2,3} are:
  (1)     (2) (3)
  (1,2)   (3)
  (1,3)   (2)
  (2,3)   (1)
  (1,2,3)
  (1,3,2)
so there are 13 elements in the first cycles, 4 in the second cycles and only 1 in the third cycles.
Triangle T(n,k) begins:
       1;
       3,     1;
      13,     4,     1;
      67,    21,     7,    1;
     411,   131,    46,   11,    1;
    2911,   950,   341,  101,   16,   1;
   23563,  7694,  2871,  932,  197,  22,  1;
  213543, 70343, 26797, 9185, 2311, 351, 29, 1;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Andrew V. Sills, Integer Partitions Probability Distributions, arXiv:1912.05306 [math.CO], 2019.
Wikipedia, Permutation

Columns k=1-10 give: A028418, A332851, A332852, A332853, A332854, A332855, A332856, A332857, A332858, A332859.
Row sums give A001563.
T(2n,n) gives A332928.
Cf. A185105, A322383.

b:= proc(n, l) option remember; `if`(n=0, add(l[-i]*
      x^i, i=1..nops(l)), add(binomial(n-1, j-1)*
      b(n-j, sort([l[], j]))*(j-1)!, j=1..n))
    end:
T:= n-> (p-> (seq(coeff(p, x, i), i=1..n)))(b(n, [])):
seq(T(n), n=1..12);

b[n_, l_] := b[n, l] = If[n == 0, Sum[l[[-i]]*x^i, {i, 1, Length[l]}], Sum[Binomial[n-1, j-1]*b[n-j, Sort[Append[l, j]]]*(j-1)!, {j, 1, n}]];
T[n_] := CoefficientList[b[n, {}], x] // Rest;
Array[T, 12] // Flatten  (* Jean-François Alcover, Feb 26 2020, after Alois P. Heinz *)

A060014 Sum of orders of all permutations of n letters.

Original entry on oeis.org

1, 1, 3, 13, 67, 471, 3271, 31333, 299223, 3291487, 39020911, 543960561, 7466726983, 118551513523, 1917378505407, 32405299019941, 608246253790591, 12219834139189263, 253767339725277823, 5591088918313739017, 126036990829657056711, 2956563745611392385211

Offset: 0

N. J. A. Sloane, Mar 17 2001

Conjecture: This sequence eventually becomes cyclic mod n for all n. - Isaac Saffold, Dec 01 2019

			For n = 4 there is 1 permutation of order 1, 9 permutations of order 2, 8 of order 3 and 6 of order 4, for a total of 67.

D. S. Mitrinovic et al., Handbook of Number Theory, Kluwer, Section XIII.2, p. 460.

Alois P. Heinz, Table of n, a(n) for n = 0..170
FindStat - Combinatorial Statistic Finder, The order of a permutation
Joshua Harrington, Lenny Jones, and Alicia Lamarche, Characterizing Finite Groups Using the Sum of the Orders of the Elements, International Journal of Combinatorics, Volume 2014, Article ID 835125, 8 pages.

Cf. A000793, A028418, A060015, A057731, A074859, A290932, A346066.

b:= proc(n, g) option remember; `if`(n=0, g, add((j-1)!
      *b(n-j, ilcm(g, j))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 1):
seq(a(n), n=0..30);  # Alois P. Heinz, Jul 11 2017

CoefficientList[Series[Sum[n Fold[#1+MoebiusMu[n/#2] Apply[Times, Exp[x^#/#]&/@Divisors[#2] ]&,0,Divisors[n]],{n,Max[Apply[LCM,Partitions[19],1]]}],{x,0,19}],x] Range[0,19]! (* Wouter Meeussen, Jun 16 2012 *)
a[ n_] := If[ n < 1, Boole[n == 0], 1 + Total @ Apply[LCM, Map[Length, First /@ PermutationCycles /@ Drop[Permutations @ Range @ n, 1], {2}], 1]]; (* Michael Somos, Aug 19 2018 *)

\\ Naive method -- sum over cycles directly
cycleDecomposition(v:vec)={
    my(cyc=List(), flag=#v+1, n);
    while((n=vecmin(v))<#v,
        my(cur=List(), i, tmp);
        while(v[i++]!=n,);
        while(v[i] != flag,
            listput(cur, tmp=v[i]);
            v[i]=flag;
            i=tmp
        );
        if(#cur>1, listput(cyc, Vec(cur)))    \\ Omit length-1 cycles
    );
    Vec(cyc)
};
permutationOrder(v:vec)={
    lcm(apply(length, cycleDecomposition(v)))
};
a(n)=sum(i=0,n!-1,permutationOrder(numtoperm(n,i)))
\\ Charles R Greathouse IV, Nov 06 2014

A060014(n) =
{
  my(factn = n!, part, nb, i, j, res = 0);
  forpart(part = n,
    nb = 1; j = 1;
    for(i = 1, #part,
      if (i == #part || part[i + 1] != part[i],
        nb *= (i + 1 - j)! * part[i]^(i + 1 - j);
        j = i + 1));
    res += (factn / nb) * lcm(Vec(part)));
  res;
} \\ Jerome Raulin, Jul 11 2017 (much faster, O(A000041(n)) vs O(n!))

E.g.f.: Sum_{n>0} (n*Sum_{i|n} (moebius(n/i)*Product_{j|i} exp(x^j/j))). - Vladeta Jovovic, Dec 29 2004; The sum over n should run to at least A000793(k) for producing the k-th entry. - Wouter Meeussen, Jun 16 2012

a(n) = Sum_{k>=1} k* A057731(n,k). - R. J. Mathar, Aug 31 2017

More terms from Vladeta Jovovic, Mar 18 2001

More terms from Alois P. Heinz, Feb 14 2013

A028417 Sum over all n! permutations of n elements of minimum lengths of cycles.

Original entry on oeis.org

1, 3, 10, 45, 236, 1505, 10914, 90601, 837304, 8610129, 96625970, 1184891081, 15665288484, 223149696601, 3394965018886, 55123430466945, 948479737691504, 17289345305870561, 332019600921360594, 6713316975465246889, 142321908843254560540, 3161718732648662557161

Offset: 1

Joe Keane (jgk(AT)jgk.org)

nonn

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

Cf. A028418, A046746, A006128.
Cf. A005225.
Column k=1 of A322383.

b:= proc(n, m) option remember; `if`(n=0, m, add((j-1)!*
      b(n-j, min(m,j))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, infinity):
seq(a(n), n=1..25);  # Alois P. Heinz, May 14 2016

Drop[Apply[Plus,Table[nn=25;Range[0,nn]!CoefficientList[Series[Exp[Sum[ x^i/i,{i,n,nn}]]-1,{x,0,nn}],x],{n,1,nn}]],1] (* Geoffrey Critzer, Jan 10 2013 *)
b[n_, m_] := b[n, m] = If[n == 0, m, Sum[(j-1)! b[n-j, Min[m, j]]* Binomial[n-1, j-1], {j, n}]];
a[n_] := b[n, Infinity];
Array[a, 25] (* Jean-François Alcover, Apr 21 2020, after Alois P. Heinz *)

E.g.f.: Sum[k>0, -1+ exp(Sum(j>=k, x^j/j))]. - Vladeta Jovovic, Jul 26 2004

a(n) = Sum_{k=1..n} k * A145877(n,k). - Alois P. Heinz, Jul 28 2014

More terms from Vladeta Jovovic, Sep 19 2002

A097147 Total sum of minimum block sizes in all partitions of n-set.

Original entry on oeis.org

1, 3, 7, 21, 66, 258, 1079, 4987, 25195, 136723, 789438, 4863268, 31693715, 217331845, 1564583770, 11795630861, 92833623206, 760811482322, 6479991883525, 57256139503047, 523919025038279, 4956976879724565, 48424420955966635, 487810283307069696

Offset: 1

Vladeta Jovovic, Jul 27 2004

Alois P. Heinz, Table of n, a(n) for n = 1..576

Cf. A028417, A028418, A046746, A006128, A097145, A097146, A097148.

Column k=1 of A319298.

g:= proc(n, i, p) option remember; `if`(n=0, (i+1)*p!,
      `if`(i<1, 0, add(g(n-i*j, i-1, p+j*i)/j!/i!^j, j=0..n/i)))
    end:
a:= n-> g(n$2, 0):
seq(a(n), n=1..30);  # Alois P. Heinz, Mar 06 2015

Drop[Apply[Plus,Table[nn=25;Range[0,nn]!CoefficientList[Series[Exp[Sum[ x^i/i!,{i,n,nn}]]-1,{x,0,nn}],x],{n,1,nn}]],1]  (* Geoffrey Critzer, Jan 10 2013 *)
g[n_, i_, p_] := g[n, i, p] = If[n == 0, (i+1)*p!, If[i<1, 0,
     Sum[g[n-i*j, i-1, p+j*i]/j!/i!^j, {j, 0, n/i}]]];
a[n_] := g[n, n, 0];
Array[a, 30] (* Jean-François Alcover, Aug 24 2021, after Alois P. Heinz *)

E.g.f.: Sum_{k>0} (-1+exp(Sum_{j>=k} x^j/j!)).

More terms from Max Alekseyev, Apr 29 2010

A097148 Total sum of maximum block sizes in all partitions of n-set.

Original entry on oeis.org

1, 3, 10, 35, 136, 577, 2682, 13435, 72310, 414761, 2524666, 16239115, 109976478, 781672543, 5814797281, 45155050875, 365223239372, 3070422740989, 26780417126048, 241927307839731, 2260138776632752, 21805163768404127, 216970086170175575, 2224040977932468379

Offset: 1

Vladeta Jovovic, Jul 27 2004

Let M be the infinite lower triangular matrix given by A080510 and v the column vector [1,2,3,...] then M*v=A097148 (this sequence, as column vector). - Gary W. Adamson, Feb 24 2011

Alois P. Heinz, Table of n, a(n) for n = 1..576

Cf. A028417, A028418, A046746, A006128, A097145-A097147.
Cf. A080510.
Column k=1 of A319375.

b:= proc(n, m) option remember; `if`(n=0, m, add(
      b(n-j, max(j, m))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=1..24);  # Alois P. Heinz, Mar 02 2020, revised May 08 2024

Drop[ Range[0, 22]! CoefficientList[ Series[ Sum[E^(E^x - 1) - E^Sum[x^j/j!, {j, 1, k}], {k, 0, 22}], {x, 0, 22}], x], 1] (* Robert G. Wilson v, Aug 05 2004 *)

E.g.f.: Sum_{k>=0} (exp(exp(x)-1)-exp(Sum_{j=1..k} x^j/j!)).

More terms from Robert G. Wilson v, Aug 05 2004

A097145 Total sum of minimum list sizes in all sets of lists of n-set, cf. A000262.

Original entry on oeis.org

0, 1, 5, 25, 157, 1101, 9211, 85513, 900033, 10402633, 133059331, 1836961941, 27619253113, 444584808253, 7678546353843, 140944884572521, 2751833492404321, 56691826303303953, 1233793951629951043, 28191548364561422173, 676190806704598883241

Offset: 0

Vladeta Jovovic, Jul 27 2004

			For n=4 we have 73 sets of lists (cf. A000262): (1234) (24 ways), (123)(4) (6*4 ways), (12)(34) (3*4 ways), (12)(3)(4) (6*2 ways), (1)(2)(3)(4) (1 way); so a(n)= 24*4+24*1+12*2+12*1+1*1 = 157.

Alois P. Heinz, Table of n, a(n) for n = 0..444

Cf. A000262, A028417, A028418, A046746, A006128, A097146-A097148.

b:= proc(n, m) option remember; `if`(n=0, m, add(j!*
      b(n-j, min(m, j))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> `if`(n=0, 0, b(n, infinity)):
seq(a(n), n=0..25);  # Alois P. Heinz, May 10 2016

b[n_, m_] := b[n, m] = If[n==0, m, Sum[j!*b[n-j, Min[m, j]]*Binomial[n-1, j - 1], {j, 1, n}]]; a[n_] := If[n==0, 0, b[n, Infinity]]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 18 2017, after Alois P. Heinz *)

E.g.f.: Sum_{k>0} (exp(x^k/(1-x))-1).

More terms from Max Alekseyev, Jul 04 2009

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

A232193 Numerators of the expected value of the length of a random cycle in a random n-permutation.

Original entry on oeis.org

1, 3, 23, 55, 1901, 4277, 198721, 16083, 14097247, 4325321, 2132509567, 4527766399, 13064406523627, 905730205, 13325653738373, 362555126427073, 14845854129333883, 57424625956493833, 333374427829017307697, 922050973293317, 236387355420350878139797

Offset: 1

Geoffrey Critzer, Nov 20 2013

In this experiment we randomly select (uniform distribution) an n-permutation and then randomly select one of the cycles from that permutation. Cf. A102928/A175441 which gives the expected cycle length when we simply randomly select a cycle.

			Expectations for n=1,... are 1/1, 3/2, 23/12, 55/24, 1901/720, 4277/1440, 198721/60480, 16083/4480, ... = A232193/A232248
For n=3 there are 6 permutations.  We have probability 1/6 of selecting (1)(2)(3) and the cycle size is 1.  We have probability 3/6 of selecting a permutation with cycle type (1)(23) and (on average) the cycle length is 3/2.  We have probability 2/6 of selecting a permutation of the form (123) and the cycle size is 3.  1/6*1 + 3/6*3/2 + 2/6*3 = 23/12.

Alois P. Heinz, Table of n, a(n) for n = 1..250

Denominators are A232248.
Cf. A028417(n)/n! the expected value of the length of the shortest cycle in a random n-permutation.
Cf. A028418(n)/n! the expected value of the length of the longest cycle in a random n-permutation.

with(combinat):
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      expand(add(multinomial(n, n-i*j, i$j)/j!*(i-1)!^j
      *b(n-i*j, i-1) *x^j, j=0..n/i))))
    end:
a:= n->numer((p->add(coeff(p, x, i)/i, i=1..n))(b(n$2))/(n-1)!):
seq(a(n), n=1..30);  # Alois P. Heinz, Nov 21 2013
# second Maple program:
a:= n-> numer(add(abs(combinat[stirling1](n, i))/i, i=1..n)/(n-1)!):
seq(a(n), n=1..30);  # Alois P. Heinz, Nov 23 2013

Table[Numerator[Total[Map[Total[#]!/Product[#[[i]],{i,1,Length[#]}]/Apply[Times,Table[Count[#,k]!,{k,1,Max[#]}]]/(Total[#]-1)!/Length[#]&,Partitions[n]]]],{n,1,25}]

a(n) = Numerator( 1/(n-1)! * Sum_{i=1..n} A132393(n,i)/i ). - Alois P. Heinz, Nov 23 2013

a(n) = numerator(Sum_{k=0..n} A002657(k)/A091137(k)) (conjectured). - Michel Marcus, Jul 19 2019

A097146 Total sum of maximum list sizes in all sets of lists of n-set, cf. A000262.

Original entry on oeis.org

0, 1, 5, 31, 217, 1781, 16501, 172915, 1998641, 25468777, 352751941, 5292123431, 85297925065, 1472161501981, 27039872306357, 527253067633531, 10865963240550241, 236088078855319505, 5390956470528548101, 129102989125943058607, 3234053809095307670201, 84596120521251178630981, 2305894874979300173268085

Offset: 0

Vladeta Jovovic, Jul 27 2004

			For n=4 we have 73 sets of lists (cf. A000262): (1234) (24 ways), (123)(4) (6*4 ways), (12)(34) (3*4 ways), (12)(3)(4) (6*2 ways), (1)(2)(3)(4) (1 way); so a(4)= 24*4+24*3+12*2+12*2+1*1 = 217.

Alois P. Heinz, Table of n, a(n) for n = 0..444

Cf. A028417, A028418, A046746, A006128, A097145, A097147, A097148.

b:= proc(n, m) option remember; `if`(n=0, m, add(j!*
      b(n-j, max(m, j))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..25);  # Alois P. Heinz, May 10 2016

b[n_, m_] := b[n, m] = If[n == 0, m, Sum[j! b[n-j, Max[m, j]] Binomial[n-1, j-1], {j, 1, n}]];
a[n_] := b[n, 0];
a /@ Range[0, 25] (* Jean-François Alcover, Nov 05 2020, after Alois P. Heinz *)

N=50; x='x+O('x^N);
egf=exp(x/(1-x))*sum(k=1,N, (1-exp(x^k/(x-1))) );
Vec( serlaplace(egf) ) /* show terms */

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

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

A208248 Sum of the maximum cycle length over all functions f:{1,2,...,n} -> {1,2,...,n} (endofunctions).

Original entry on oeis.org

0, 1, 5, 40, 431, 5826, 94657, 1795900, 38963535, 951398890, 25819760021, 770959012704, 25117397416795, 886626537549130, 33708625339151505, 1373237757290215156, 59677939242566840303, 2755753623830236494930, 134746033233724391374765, 6954962673986411576581000

Offset: 0

Geoffrey Critzer, Jan 12 2013

nonn

a(n) is also the sum of the number of endofunctions with at least one cycle >= i for all i >= 1. In other words, a(n) = A000312(n) + A101334(n) + A208240(n) + ... .

Alois P. Heinz, Table of n, a(n) for n = 0..386

Cf. A000169, A028418, A000312, A101334, A208231, A208240, A290932.

b:= proc(n, m) option remember; `if`(n=0, m, add((j-1)!*
      b(n-j, max(m, j))*binomial(n-1, j-1), j=1..n))
    end:
a:= n-> add(b(j, 0)*n^(n-j)*binomial(n-1, j-1), j=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, May 20 2016

nn=20; t=Sum[n^(n-1)x^n/n!, {n,1,nn}]; Apply[Plus, Table[Range[0,nn]! CoefficientList[Series[1/(1-t) - Exp[Sum[t^i/i, {i,1,n}]], {x,0,nn}], x], {n, 0, nn-1}]]

E.g.f.: Sum_{k>=0} 1/(1-T(x)) - exp(Sum_{i=1...k} T(x)^i/i) = A(T(x)) where A(x) is the e.g.f. for A028418 and T(x) is the e.g.f. for A000169.

cp's OEIS Frontend

A126074 Triangle read by rows: T(n,k) is the number of permutations of n elements that have the longest cycle length k.

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A322384 Number T(n,k) of entries in the k-th cycles of all permutations of [n] when cycles are ordered by decreasing lengths (and increasing smallest elements); triangle T(n,k), n>=1, 1<=k<=n, read by rows.

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A060014 Sum of orders of all permutations of n letters.

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PARI

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A028417 Sum over all n! permutations of n elements of minimum lengths of cycles.

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A097147 Total sum of minimum block sizes in all partitions of n-set.

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A097148 Total sum of maximum block sizes in all partitions of n-set.

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A097145 Total sum of minimum list sizes in all sets of lists of n-set, cf. A000262.

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