

A046682


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


39



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.
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
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)


LINKS



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


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. *)
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 *)


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

For the number of conjugacy classes of the alternating group A_n, n>=2, see A000702.
Heinz number (rank) and partition:
 A352488 = rank greater than or equal to conjugate, counted by A046682.


KEYWORD

nonn,nice


AUTHOR



STATUS

approved



