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Self-Conjugate Partition


SelfConjugatePartitions

A partition whose conjugate partition is equivalent to itself. The Ferrers diagrams corresponding to the self-conjugate partitions for 3<=n<=10 are illustrated above. The numbers of self-conjugate partitions of n=1, 2, ... are 1, 0, 1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 4, 5, 5, 5, 6, 7, ... (OEIS A000700).

The number of self-conjugate partitions S_n of n is equal to the number of partitions of n into distinct odd parts, and has generating function

product_(k=0)^(infty)1+x^(2k+1)=sum_(k=0)^(infty)S_kx^k
(1)
=(-x;x^2)_infty
(2)
=1+x+x^3+x^4+x^5+x^6+x^7+2x^8+2x^9+...,
(3)

and (-1)^nS_n has generating function

product_(k=1)^(infty)1/(1+x^k)=sum_(k=0)^(infty)(-1)^kS_kx^k
(4)
=2/((1;x)_infty)
(5)
=1-x-x^3+x^4-x^5+x^6-x^7+2x^8-2x^9+...,
(6)

where (q;a)_infty is a q-Pochhammer symbol.


See also

Conjugate 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. Oxford, England: Clarendon Press, p. 277, 1979.Osima, M. "On the Irreducible Representations of the Symmetric Group." Canad. J. Math. 4, 381-384, 1952.Watson, G. N. "Two Tables of Partitions." Proc. London Math. Soc. 42, 550-556, 1936.

Referenced on Wolfram|Alpha

Self-Conjugate Partition

Cite this as:

Weisstein, Eric W. "Self-Conjugate Partition." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Self-ConjugatePartition.html

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