Restricted inversion sequences and enhanced $3$-noncrossing partitions

We prove a conjecture due independently to Yan and Martinez--Savage that asserts inversion sequences with no weakly decreasing subsequence of length $3$ and enhanced $3$-noncrossing partitions have the same cardinality. Our approach applies both the generating tree technique and the so-called obstinate kernel method developed by Bousquet-M\'elou. One application of this equinumerosity is a discovery of an intriguing identity involving numbers of classical and enhanced $3$-noncrossing partitions.