

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
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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*n1) = A027193(2*n1); 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), 381384.
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} (1q^m ) = ( 1/Product_{m>=1} (1q^m) + Product_{m>=1} (1+q^(2*m1) ) ) / 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)^(nk)*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[1q^n, {n, 1, max}]; CoefficientList[ Series[f[q], {q, 0, max}], q] (* JeanFrançois 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, 1q^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



