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A bijective proof of the Delannoy recurrence. (English) Zbl 1030.05003
Summary: The central Delannoy numbers $1, 3, 13, 63, 321, 1683,\dots$ satisfy the recurrence $$\align nd_n &= 3(2n- 1)d_{n- 1}- (n- 1)d_{n- 2};\quad n\ge 2;\quad d_0= 1,\quad d_1= 3,\\ d_n &= [x^n]\Biggl\{{1\over \sqrt{x^2- 6x+ 1}}\Biggr\}.\endalign$$ In this paper, we provide a bijective proof of the above recurrence. It does not appear that there is any previous combinatorial proof of the Delannoy recurrence. In our proof, we use lattice paths of length $2n$ from $(0,0)$ to $(2n,0)$ with possible steps $(1,1)$, $(1,-1)$, and $(2,0)$. If $D_n$ is the set of all such paths, then the cardinality of $D_n$ is $d_n$, the $n$th Delannoy number. Let $S_n$ be the subset of paths in $D_n$ which do not go below the x-axis. These paths are called Schröder paths and the cardinality of $S_n$ is the $n$th large Schröder number $s_n$. The large Schröder numbers $1, 2, 6, 22, 90, 394,\dots$ satisfy the recurrence $$\align (n+1)s_n &= 3(2n- 1)s_{n- 1}-(n- 2)s_{n-2};\quad n\ge 2;\quad s_0= 1,\quad s_1= 2,\\ s_n &= [x^n]\Biggl\{{1- x- \sqrt{x^2- 6x+1}\over 2x}\Biggr\}.\endalign$$ We show that the Delannoy recurrence can be interpreted bijectively in the same manner as the Schröder recurrence.

MSC:
05A10Combinatorial functions
05A15Exact enumeration problems, generating functions