An area-bounce exchanging bijection on a large subset of Dyck paths
It is a longstanding open problem to find a bijection exchanging area and bounce statistics on Dyck paths. We settle this problem for an exponentially large subset of Dyck paths via an explicit bijection. Moreover, we prove that this bijection is natural by showing that it maps what we call bounce-m...
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Zusammenfassung: | It is a longstanding open problem to find a bijection exchanging area and
bounce statistics on Dyck paths. We settle this problem for an exponentially
large subset of Dyck paths via an explicit bijection. Moreover, we prove that
this bijection is natural by showing that it maps what we call bounce-minimal
paths to area-minimal paths. As a consequence of the proof ideas, we show
combinatorially that a path with area $a$ and bounce $b$ exists if and only if
a path with area $b$ and bounce $a$ exists. We finally show that the number of
distinct values of the sum of the area and bounce statistics is the number of
nonzero coefficients in Johnson's $q$-Bell polynomial. |
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DOI: | 10.48550/arxiv.2401.14668 |