Abstract
We consider a class of lattice paths with certain restrictions on their ascents and down-steps and use them as building blocks to construct various families of Dyck paths. We let every building block \(P_j\) take on \(c_j\) colors and count all of the resulting colored Dyck paths of a given semilength. Our approach is to prove a recurrence relation of convolution type, which yields a representation in terms of partial Bell polynomials that simplifies the handling of different colorings. This allows us to recover multiple known formulas for Dyck paths and related lattice paths in a unified manner.
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Notes
- 1.
A word in a language L is said to be factor-free if it has no proper factor in L.
- 2.
For more on this bijection for general rational Dyck paths, we refer to [3].
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Birmajer, D., Gil, J.B., McNamara, P.R.W., Weiner, M.D. (2019). Enumeration of Colored Dyck Paths Via Partial Bell Polynomials. In: Andrews, G., Krattenthaler, C., Krinik, A. (eds) Lattice Path Combinatorics and Applications. Developments in Mathematics, vol 58. Springer, Cham. https://doi.org/10.1007/978-3-030-11102-1_8
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