Abstract
A small q-Schroder path of semilength n is a lattice path from (0, 0) to (2n, 0) using up steps $$U = (1, 1)$$ , horizontal steps $$H = (2, 0)$$ , and down steps $$D = (1,-1)$$ such that it stays weakly above the x-axis, has no horizontal steps on the x-axis, and each horizontal step comes in q colors. In this paper, we provide a bijection between the set of small q-Schroder paths of semilength $$n+1$$ and the set of $$(q+2, q+1)$$ -Motzkin paths of length n. Furthermore, a one-to-one correspondence between the set of small 3-Schroder paths of semilength n and the set of Catalan rook paths of semilength n is obtained, and a bijection between small 4-Schroder paths and Catalan queen paths is also presented.
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