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A046682 Number of cycle types of conjugacy classes of all even permutations of n elements. 26
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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

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.

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.

EXAMPLE

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....

  ..... ..... ..... .....

  ..... ..... ..... .....

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. *)

PROG

(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 */

CROSSREFS

Cf. A000041, A000700, A000701, A006950, A015128.

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

Cf. A118301.

Sequence in context: A241743 A321729 A180652 * A005987 A241828 A125895

Adjacent sequences:  A046679 A046680 A046681 * A046683 A046684 A046685

KEYWORD

nonn,nice

AUTHOR

Vladeta Jovovic

STATUS

approved

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Last modified October 20 19:08 EDT 2020. Contains 337906 sequences. (Running on oeis4.)