A046682 Number of cycle types of conjugacy classes of all even permutations of n elements.

1, 1, 1, 2, 3, 4, 6, 8, 12, 16, 22, 29, 40, 52, 69, 90, 118, 151, 195, 248, 317, 400, 505, 632, 793, 985, 1224, 1512, 1867, 2291, 2811, 3431, 4186, 5084, 6168, 7456, 9005, 10836, 13026, 15613, 18692, 22316, 26613, 31659, 37619, 44601, 52815, 62416, 73680, 86809, 102162

Offset: 0

Author

Vladeta Jovovic

Keywords

• nonn
• nice

Comments

Also number of partitions of n with even number of even parts. There is no restriction on the odd parts.

a(n) = u(n) + v(n), n >= 2, of the Osima reference, p. 383.

Also number of partitions of n with largest part congruent to n modulo 2: a(2*n) = A027187(2*n), a(2*n-1) = A027193(2*n-1); a(n) = A000041(n) - A000701(n). - Reinhard Zumkeller, Apr 22 2006

Equivalently, number of partitions of n with number of parts having the same parity as n. - Olivier Gérard, Apr 04 2012

Also number of distinct free Young diagrams (Ferrers graphs with n nodes). Free Young diagrams are distinct when none is a rigid transformation (translation, rotation, reflection or glide reflection) of another. - Jani Melik, May 08 2016

Let the cycle type of an even permutation be represented by the partition A=(O1,O2,...,Oi,E1,E2,...,E2j), where the Os are parts with odd length and the Es are parts with even lengths, and where j may be zero, using Reinhard Zumkeller's observation that the partition associated with a cycle type of an even permutation has an even number of even parts. The set of even cycle types enumerated here can be considered a monoid under a binary operation *: Let A be as above and B=(o1,o2,...,ok,e1,e2,...,e2m). A*B is the partition (O1o1,O1o2,...,O1ok,O1e1,...,O1e2m,O2o1,...,O2e2m,...,Oio1,...,Oie2m,E1o1,...,E1e2m,...,E2je2m). This product has 2im+2jk+4jm even parts, so it represents the cycle type of an even permutation. - Richard Locke Peterson, Aug 20 2018

From Gus Wiseman, Mar 31 2022: (Start)

Also the number of integer partitions of n with Heinz number greater than or equal to that of their conjugate, where the Heinz number of a partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). These partitions are ranked by A352488. The complement is counted by A000701. For example, the a(n) partitions for n = 1...7 are:

(1) (11) (21) (22) (221) (222) (331)

(111) (211) (311) (321) (2221)

(1111) (2111) (2211) (3211)

(11111) (3111) (4111)

(21111) (22111)

(111111) (31111)

(211111)

(1111111)

Also the number of integer partitions of n with Heinz number less than or equal to their conjugate, ranked by A352489. For example, the a(n) partitions for n = 1...7 are:

(1) (2) (3) (4) (5) (6) (7)

(21) (22) (32) (33) (43)

(31) (41) (42) (52)

(311) (51) (61)

(321) (322)

(411) (421)

(511)

(4111)

(End)

Examples

			1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 12*x^8 + 16*x^9 + ...
a(3)=2 since cycle types of even permutations of 3 elements is (.)(.)(.), (...).
a(4)=3 since cycle types of even permutations of 4 elements is (.)(.)(.)(.), (...)(.), (..)(..).
a(5)=4 (free Young diagrams):
  XXXXX XXXX. XXX.. XXX..
  ..... X.... XX... X....
  ..... ..... ..... X....
  ..... ..... ..... .....
  ..... ..... ..... .....
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
• George E. Andrews, David Newman, The Minimal Excludant in Integer Partitions, J. Int. Seq., Vol. 23 (2020), Article 20.2.3.
• J. Huh and B. Kim, The number of equivalence classes arising from partition involutions, Int. J. Number Theory, 16 (2020), 925-939.
• M. Osima, On the irreducible representations of the symmetric group, Canad. J. Math., 4 (1952), 381-384.
• Sheila Sundaram, On a positivity conjecture in the character table of S_n, arXiv:1808.01416 [math.CO], 2018.

Crossrefs

Cf. A000701, A006950, A015128.

For the number of conjugacy classes of the alternating group A_n, n>=2, see A000702.

Cf. A118301.

A000041 counts integer partitions.

A000700 counts self-conjugate partitions, ranked by A088902.

A330644 counts non-self-conjugate partitions, ranked by A352486.

Heinz number (rank) and partition:

- A122111 = rank of conjugate.

- A296150 = parts of partition, conjugate A321649.

- A352487 = rank less than conjugate, counted by A000701.

- A352488 = rank greater than or equal to conjugate, counted by A046682.

- A352489 = rank less than or equal to conjugate, counted by A046682.

- A352490 = rank greater than conjugate, counted by A000701.

- A352491 = rank minus conjugate.

Cf. A114088, A115994, A171966, A238352, A258116, A321648, A325039.

Programs

Maple

seq(add((-1)^(n-k)*combinat:-numbpart(n,k),k=0..n),n=0..48); # Peter Luschny, Aug 03 2015

Mathematica

max = 48; f[q_] := Sum[(-q^2)^n^2, {n, 0, max}]/Product[1-q^n, {n, 1, max}]; CoefficientList[ Series[f[q], {q, 0, max}], q] (* Jean-François Alcover, Oct 18 2011, after g.f. *)
conj[y_]:=If[Length[y]==0,y,Table[Length[Select[y,#>=k&]],{k,1,Max[y]}]];
Table[Length[Select[IntegerPartitions[n],Times@@Prime/@#>=Times@@Prime/@conj[#]&]],{n,0,15}] (* Gus Wiseman, Mar 31 2022 *)

PARI

list(lim)=my(q='q);Vec(sum(n=0,sqrt(lim),(-q^2)^(n^2))/prod(n=1,lim,1-q^n)+O(q^(lim\1+1))) \\ Charles R Greathouse IV, Oct 18 2011

PARI

{a(n) = if( n<0, 0, (numbpart(n) + polcoeff( 1 / prod( k=1, n, 1 + (-x)^k, 1 + x * O(x^n)), n)) / 2)} /* Michael Somos, Jul 24 2012 */

Formula

G.f.: Sum_{n>=0} (-q^2)^(n^2) / Product_{m>=1} (1-q^m ) = ( 1/Product_{m>=1} (1-q^m) + Product_{m>=1} (1+q^(2*m-1) ) ) / 2. - Mamuka Jibladze, Sep 07 2003

a(n) = (A000041(n) + A000700(n)) / 2.

a(n) = A000041(n) - A000701(n). - Gus Wiseman, Mar 31 2022