A self-conjugate partition is a partition whose conjugate partition is equal to itself. The Ferrers diagrams corresponding to the self-conjugate
partitions for A000700 ).
The number nonnegative
integer partitions of odd parts, with
partition . Its generating
function is
and generating function
where q -Pochhammer symbolformal power series and for
See also Conjugate Partition ,
Core Partition ,
Ferrers Diagram ,
Partition
Function P
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References Hardy, G. H. and Wright, E. M. An Introduction to the Theory of Numbers, 5th ed. Osima, M. "On the Irreducible Representations
of the Symmetric Group." Canad. J. Math. 4 , 381-384, 1952. https://doi.org/10.4153/cjm-1952-034-x . Sloane,
N. J. A. Sequence A000700 in "The
On-Line Encyclopedia of Integer Sequences." Watson, G. N. "Two
Tables of Partitions." Proc. London Math. Soc. 42 , 550-556, 1937.
https://doi.org/10.1112/plms/s2-42.1.550 . Referenced
on Wolfram|Alpha Self-Conjugate Partition
Cite this as:
Weisstein, Eric W. "Self-Conjugate Partition."
From MathWorld https://mathworld.wolfram.com/Self-ConjugatePartition.html
Subject classifications
are illustrated above. The numbers of self-conjugate partitions of 
, 2, ... are 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4,
5, 5, 5, 6, 7, ... (OEIS A000700).
of self-conjugate partitions of a nonnegative
integer 
equals the number of partitions of 
into distinct odd parts, with
for the empty partition. Its generating
function is
has generating function
is a q-Pochhammer symbol. The identities
hold as formal power series and for 
.
