Dyck paths on colored lattices

Abstract

Fried recently enumerated Dyck paths having equally many black and white cells below them, for the chessboard coloring (Narayana numbers) and the column-alternating coloring (Fuss--Catalan numbers). We prove a generalization here: for the coloring of columns modulo any $c\ge2$, the number of Dyck paths of semilength $n$ whose $c$ residue classes carry equal weight is the Raney number $\Raney_{c+1,r}(m)$, where $n=cm+r-1$.

Disclosure

“2020 Mathematics Subject Classification. 05A15, 05A19. Key words and phrases. Dyck paths, Catalan numbers, Raney numbers, cycle lemma. Claude Opus 4.8 and Claude Sonnet 5 were used to edit the note, and make notations more efficient. The author takes responsibility for all of the mathematical content.”

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2026-08-24T00:34:15.228Z

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