An involution on lattice paths between two boundaries
نویسنده
چکیده
We give an involution on the set of lattice paths from (0, 0) to (a, b) with steps N = (0, 1) and E = (1, 0) that lie between two boundaries T and B, which proves that the statistics ‘number of E steps shared with T ’ and ‘number of E steps shared with B’ have a symmetric joint distribution on this set. This generalizes a result of Deutsch for the case of Dyck paths.
منابع مشابه
Two-boundary lattice paths and parking functions
We describe an involution on a set of sequences associated with lattice paths with north or east steps constrained to lie between two arbitrary boundaries. This involution yields recursions (from which determinantal formulas can be derived) for the number and area enumerator of such paths. An analogous involution can be defined for parking functions with arbitrary lower and upper bounds. From t...
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We prove that on the set of lattice paths with steps N = (0, 1) and E = (1, 0) that lie between two boundaries B and T , the two statistics ‘number of E steps shared with B’ and ‘number of E steps shared with T ’ have a symmetric joint distribution. We give an involution that switches these statistics, preserves additional parameters, and generalizes to paths that contain steps S = (0,−1) at pr...
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