TY - JOUR
T1 - Hook length biases and general linear partition inequalities
AU - Ballantine, Cristina
AU - Burson, Hannah E.
AU - Craig, William
AU - Folsom, Amanda
AU - Wen, Boya
N1 - Publisher Copyright:
© 2023, The Author(s), under exclusive licence to Springer Nature Switzerland AG.
PY - 2023/12
Y1 - 2023/12
N2 - Motivated in part by hook-content formulas for certain restricted partitions in representation theory, we consider the total number of hooks of fixed length in odd versus distinct partitions. We show that there are more hooks of length 2, respectively 3, in all odd partitions of n than in all distinct partitions of n, and make the analogous conjecture for arbitrary hook length t≥ 2 . We also establish additional bias results on the number of gaps of size 1, respectively 2, in all odd versus distinct partitions of n. We conjecture similar biases and asymptotics, as well as congruences for the number of hooks of fixed length in odd distinct partitions versus self-conjugate partitions. An integral component of the proof of our bias result for hooks of length 3 is a linear inequality involving q(n), the number of distinct partitions of n. In this article we also establish effective linear inequalities for q(n) in great generality, a result which is of independent interest. Our methods are both analytic and combinatorial, and our results and conjectures intersect the areas of representation theory, analytic number theory, partition theory, and q-series. In particular, we use a Rademacher-type exact formula for q(n), Wright’s circle method, modularity, q-series transformations, asymptotic methods, and combinatorial arguments.
AB - Motivated in part by hook-content formulas for certain restricted partitions in representation theory, we consider the total number of hooks of fixed length in odd versus distinct partitions. We show that there are more hooks of length 2, respectively 3, in all odd partitions of n than in all distinct partitions of n, and make the analogous conjecture for arbitrary hook length t≥ 2 . We also establish additional bias results on the number of gaps of size 1, respectively 2, in all odd versus distinct partitions of n. We conjecture similar biases and asymptotics, as well as congruences for the number of hooks of fixed length in odd distinct partitions versus self-conjugate partitions. An integral component of the proof of our bias result for hooks of length 3 is a linear inequality involving q(n), the number of distinct partitions of n. In this article we also establish effective linear inequalities for q(n) in great generality, a result which is of independent interest. Our methods are both analytic and combinatorial, and our results and conjectures intersect the areas of representation theory, analytic number theory, partition theory, and q-series. In particular, we use a Rademacher-type exact formula for q(n), Wright’s circle method, modularity, q-series transformations, asymptotic methods, and combinatorial arguments.
KW - Distinct partitions
KW - Hook length
KW - Odd partitions
KW - Partition asymptotics
KW - Partition inequalities
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U2 - 10.1007/s40687-023-00402-1
DO - 10.1007/s40687-023-00402-1
M3 - Article
AN - SCOPUS:85173950980
SN - 2522-0144
VL - 10
JO - Research in Mathematical Sciences
JF - Research in Mathematical Sciences
IS - 4
M1 - 41
ER -