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

0,4

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

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)

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.

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

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. *)
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, 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. 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.

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