cp's OEIS Frontend

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

%I A059171 #49 Feb 01 2025 23:19:47
%S A059171 1,1,3,8,30,144,840,5760,45360,403200,3991680,43545600,518918400,
%T A059171 6706022400,93405312000,1394852659200,22230464256000,376610217984000,
%U A059171 6758061133824000,128047474114560000,2554547108585472000,53523844179886080000,1175091669949317120000
%N A059171 Size of largest conjugacy class in S_n, the symmetric group on n symbols.
%C A059171 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
%H A059171 Vincenzo Librandi, <a href="/A059171/b059171.txt">Table of n, a(n) for n = 1..200</a>
%H A059171 Jun Yan, <a href="https://arxiv.org/abs/2404.07958">Results on pattern avoidance in parking functions</a>, arXiv:2404.07958 [math.CO], 2024. See p. 4.
%F A059171 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.
%F A059171 E.g.f.: -log(1-x) - x + 1/(1-x). [for a(n+1) - _Michael Somos_, Aug 26 2015]
%F A059171 E.g.f.: x - x^2/2 - x*log(1-x). - _Michael Somos_, Aug 26 2015
%F A059171 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
%F A059171 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
%F A059171 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
%F A059171 From _Amiram Eldar_, Jan 22 2023: (Start)
%F A059171 Sum_{n>=1} 1/a(n) = 5/2.
%F A059171 Sum_{n>=1} (-1)^(n+1)/a(n) = 2/e - 1/2. (End)
%e A059171 a(3) = 3 because the largest conjugacy class in S_3 consists of the three 2-cycles {(12),(13),(23)}.
%e A059171 G.f. = x + x^2 + 3*x^3 + 8*x^4 + 30*x^5 + 144*x^6 + 840*x^7 + 5760*x^8 + ...
%p A059171 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:
%t A059171 Join[{1,1},Table[n (n-2)!,{n,3,30}]] (* _Harvey P. Dale_, Oct 25 2011 *)
%t A059171 a[ n_] := If[ n < 0, 0, n! SeriesCoefficient[ x - x^2/2 - x Log[1 - x], {x, 0, n}]]; (* _Michael Somos_, Aug 26 2015 *)
%t A059171 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 *)
%o A059171 (Magma) [1, 1] cat [n*Factorial(n-2): n in [3..25]]; // _Vincenzo Librandi_, Oct 26 2011
%o A059171 (PARI) Vec(1+x*serlaplace((1+x-x^2)/(1-x)^2+O(x^66))) \\ _Joerg Arndt_, Mar 28 2013
%o A059171 (PARI) a(n)=if(n<=1,1,n!/(n-1)); \\ _Joerg Arndt_, Mar 28 2013
%Y A059171 Cf. A001048, A065475, A181897.
%K A059171 nonn,easy,nice
%O A059171 1,3
%A A059171 _Des MacHale_, Feb 14 2001
%E A059171 More terms from Larry Reeves (larryr(AT)acm.org), _Fabian Rothelius_ and James A. Sellers, Feb 15 2001