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-Mélou. One application of this equinumerosity is a discovery of an intriguing identity involving numbers of classical and enhanced 3-noncrossing partitions.