Combinatorial Proofs of Addition Formulas

  • Xiang-Ke Chang
  • Xing-Biao Hu
  • Hongchuan Lei
  • Yeong-Nan Yeh
Keywords: Lattice paths, Catalan numbers, Motzkin numbers, Large Schr\"{o}der numbers, Hankel determinant


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.

Author Biography

Xiang-Ke Chang
LSEC, ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences
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