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A046682 Number of cycle types of conjugacy classes of all even permutations of n elements. 25
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)

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 December 5 21:00 EST 2019. Contains 329779 sequences. (Running on oeis4.)