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We study heaps of pieces for lattice paths, which give a combinatorialvisualization of lattice paths. We introduce two types of heaps: type $I$ andtype $II$. A heap of type $I$ is characterized by peaks of a lattice path. Wehave a duality between a lattice path $\mu$ and its dual $\overline{\mu}$ onheaps of type $I$. A heap of type $II$ for $\mu$ is characterized by the skewshape between the lowest path and $\mu$. We give a determinant expression forthe generating function of heaps for general lattice paths, and an explicitformula for rational $(1,k)$-Dyck paths by using the inversion lemma. Weintroduce and study heaps in $k+1$-dimensions which are bijective to heaps oftype $II$ for $(1,k)$-Dyck paths. Further, we show a bijective correspondencebetween type $I$ and type $II$ in the case of rational $(1,k)$-Dyck paths. Asanother application of heaps, we give two explicit formulae for the generatingfunction of heaps for symmetric Dyck paths in terms of statistics on Dyck pathsand on symmetric Dyck paths respectively.

Heaps of pieces for lattice paths