TY - JOUR
T1 - Almost partition identities
AU - Andrews, George E.
AU - Ballantine, Cristina
N1 - Publisher Copyright:
© 2019 National Academy of Sciences. All Rights Reserved.
PY - 2019
Y1 - 2019
N2 - An almost partition identity is an identity for partition numbers that is true asymptotically 100% of the time and fails infinitely often. We prove a kind of almost partition identity, namely that the number of parts in all self-conjugate partitions of n is almost always equal to the number of partitions of n in which no odd part is repeated and there is exactly one even part (possibly repeated). Not only does the identity fail infinitely often, but also, the error grows without bound. In addition, we prove several identities involving the number of parts in restricted partitions. We show that the difference in the number of parts in all self-conjugate partitions of n and the number of parts in all partitions of n into distinct odd parts equals the number of partitions of n in which no odd part is repeated, the smallest part is odd, and there is exactly one even part (possibly repeated). We provide both analytic and combinatorial proofs of this identity.
AB - An almost partition identity is an identity for partition numbers that is true asymptotically 100% of the time and fails infinitely often. We prove a kind of almost partition identity, namely that the number of parts in all self-conjugate partitions of n is almost always equal to the number of partitions of n in which no odd part is repeated and there is exactly one even part (possibly repeated). Not only does the identity fail infinitely often, but also, the error grows without bound. In addition, we prove several identities involving the number of parts in restricted partitions. We show that the difference in the number of parts in all self-conjugate partitions of n and the number of parts in all partitions of n into distinct odd parts equals the number of partitions of n in which no odd part is repeated, the smallest part is odd, and there is exactly one even part (possibly repeated). We provide both analytic and combinatorial proofs of this identity.
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U2 - 10.1073/pnas.1820945116
DO - 10.1073/pnas.1820945116
M3 - Article
C2 - 30833382
AN - SCOPUS:85063278367
SN - 0027-8424
VL - 116
SP - 5428
EP - 5436
JO - Proceedings of the National Academy of Sciences of the United States of America
JF - Proceedings of the National Academy of Sciences of the United States of America
IS - 12
ER -