On an Involution on Partitions with Crank 0
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Kaavya introduce an involution on the set of partitions with crank 0 and studied the number of partitions of n which are invariant under Kaavya’s involution. If a partition λ with crank 0 is invariant under her involution, we say λ is a self- conjugate partition with crank 0. We prove that the number of such partitions of n is equal to the number of partitions with rank 0 which are invariant under the usual partition conjugation. We also study arithmetic properties of such partitions and their q-theoretic implication.
Abstract
Kaavya introduce an involution on the set of partitions with crank 0 and studied the number of partitions of n which are invariant under Kaavya’s involution. If a partition λ with crank 0 is invariant under her involution, we say λ is a self- conjugate partition with crank 0. We prove that the number of such partitions of n is equal to the number of partitions with rank 0 which are invariant under the usual partition conjugation. We also study arithmetic properties of such partitions and their q-theoretic implication.
참고문헌 (7)
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- F. Garvan, "New combinatorial interpretations of Ramanujan's partition congruences mod 5, 7, and 11", Trans. Amer. Math. Soc, 305, pp.47~77 (1988)
- S. J. Kaavya, "Crank 0 partitions and the parity of the partition function", Int. J. Number Theory, 7, pp.793~801 (2011)