The partition rank of a nonempty partition is its largest part minus its number of parts (Dyson 1944). For example, the partition 
of 9 has rank 
. Taking the conjugate
partition interchanges the largest part and the number of parts, so it negates
the rank. Dyson introduced this statistic to account for congruences
of the partition function P(n)
modulo 5 and 7.
Let 
count the partitions of 
with rank 
. Mao et al. (2026) proved
that the sequence 
indexed by 
is a logarithmically concave sequence
for 
, with strict inequality 
at every interior index 
.
