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- W2893598779 abstract "Irving and Rattan gave a formula for counting lattice paths dominated by a cyclically shifting piecewise linear boundary of varying slope. Their main result may be considered as a deep extension of well-known enumerative formulas concerning lattice paths from (0, 0) to (kn, n) lying under the line $$x=ky$$ (e.g., the Dyck paths when $$k=1$$ ). On the other hand, the classical Chung–Feller theorem tells us that the number of lattice paths from (0, 0) to (n, n) with exactly 2k steps above the line $$x=y$$ is independent of k and is therefore the Catalan number $$frac{1}{n+1}{2natopwithdelims ()n}$$ . In this paper, we study the number of lattice path boundary pairs $$(P,mathbf{a})$$ with k flaws, where P is a lattice path from (0, 0) to (n, m), $$mathbf{a}$$ is a weak m-part composition of n, and a flaw is a horizontal step of P above the boundary $$partial mathbf{a}$$ . We prove bijectively, for a given $$mathbf{a}$$ , that summing these numbers over all cyclic shifts of the boundary $$partial mathbf{a}$$ is equal to $${n+matopwithdelims ()m-1}$$ . That is, we generalize the Irving–Rattan formula to a Chung–Feller-type theorem. We also give a refinement of this result by taking the number of double ascents of lattice paths into account." @default.
- W2893598779 created "2018-10-05" @default.
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- W2893598779 date "2018-10-01" @default.
- W2893598779 modified "2023-09-25" @default.
- W2893598779 title "A Chung–Feller theorem for lattice paths with respect to cyclically shifting boundaries" @default.
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- W2893598779 doi "https://doi.org/10.1007/s10801-018-0845-z" @default.
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