A059171 Size of largest conjugacy class in S_n, the symmetric group on n symbols.

1, 1, 3, 8, 30, 144, 840, 5760, 45360, 403200, 3991680, 43545600, 518918400, 6706022400, 93405312000, 1394852659200, 22230464256000, 376610217984000, 6758061133824000, 128047474114560000, 2554547108585472000, 53523844179886080000, 1175091669949317120000

Offset

1,3

Comments

Apart from initial terms, same as A001048. The number a(n) is the maximum of row n in the triangle of refined rencontres numbers A181897. - Tilman Piesk, Apr 02 2012

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Jun Yan, Results on pattern avoidance in parking functions, arXiv:2404.07958 [math.CO], 2024. See p. 4.

Formula

a(1) = a(2) = 1; a(n) = n*(n-2)! = (n!)/(n-1) for n > 2. This is the number of (n-1)-cycles in S_n.

E.g.f.: -log(1-x) - x + 1/(1-x). [for a(n+1) - Michael Somos, Aug 26 2015]

E.g.f.: x - x^2/2 - x*log(1-x). - Michael Somos, Aug 26 2015

The sequence 1, 3, 8, ... has e.g.f. (1+x-x^2)/(1-x)^2 and a(n) = n!(n+2-0^n) = n!*A065475(n). - Paul Barry, May 14 2004

E.g.f.: E(0) - x, where E(k) = 1 + x/(k+1)/(1 - 1/(1 + 1/(k+1)/E(k+1))); (continued fraction). - Sergei N. Gladkovskii, Mar 27 2013

G.f.: 1 + x/Q(0), where Q(k)= 1 - x/(1+x) - x/(1+x)*(k+2)/(1 - x/(1+x)*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 22 2013

From Amiram Eldar, Jan 22 2023: (Start)

Sum_{n>=1} 1/a(n) = 5/2.

Sum_{n>=1} (-1)^(n+1)/a(n) = 2/e - 1/2. (End)

Example

a(3) = 3 because the largest conjugacy class in S_3 consists of the three 2-cycles {(12),(13),(23)}.
G.f. = x + x^2 + 3*x^3 + 8*x^4 + 30*x^5 + 144*x^6 + 840*x^7 + 5760*x^8 + ...

Maple

a := proc(n) if n<=2 then RETURN(1) else RETURN(n*(n-2)!) fi: end:for n from 1 to 40 do printf(`%d, `, a(n)) od:

Mathematica

Join[{1, 1}, Table[n (n-2)!, {n, 3, 30}]] (* Harvey P. Dale, Oct 25 2011 *)
a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ x - x^2/2 - x Log[1 - x], {x, 0, n}]]; (* Michael Somos, Aug 26 2015 *)
a[ n_] := With[ {m = n - 1}, If[ m < 0, 0, m! SeriesCoefficient[ -Log[1 - x] - x + 1/(1 - x), {x, 0, m}]]]; (* Michael Somos, Aug 26 2015 *)

Prog

(Magma) [1, 1], [n*Factorial(n-2): n in [3..25]]; // Vincenzo Librandi, Oct 26 2011
(PARI) Vec(1+x*serlaplace((1+x-x^2)/(1-x)^2+O(x^66))) \\ Joerg Arndt, Mar 28 2013
(PARI) a(n)=if(n<=1, 1, n!/(n-1)); \\ Joerg Arndt, Mar 28 2013

Crossrefs

Cf. A001048, A065475, A181897.

Sequence in context: A009123 A066764 A349426 * A261766 A078619 A274934

Adjacent sequences: A059168 A059169 A059170 * A059172 A059173 A059174

Keyword

nonn,easy,nice

Author

Des MacHale, Feb 14 2001

Extensions

More terms from Larry Reeves (larryr(AT)acm.org), Fabian Rothelius and James A. Sellers, Feb 15 2001

Status

approved