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

Restricted inversion sequences and enhanced $3$-noncrossing partitions