Combinatorial Proofs of Addition Formulas

Xiang-Ke Chang, Xing-Biao Hu, Hongchuan Lei, Yeong-Nan Yeh


In this paper we give a combinatorial proof of an addition formula for weighted partial Motzkin paths. The addition formula allows us to determine the $LDU$ decomposition of a Hankel matrix of the polynomial sequence defined by weighted partial Motzkin paths. As a direct consequence, we get the determinant of the Hankel matrix of certain combinatorial sequences. In addition, we obtain an addition formula for weighted large Schröder paths.


Lattice paths; Catalan numbers; Motzkin numbers; Large Schr\"{o}der numbers; Hankel determinant

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