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Give a bijective proof-----the number of oriented increasing binary trees on the vertex set $\{$ $1,2,\cdots,n$ $\}$$\lbrace 1,2,\cdots,n\rbrace$ is the Eulerian number $E_n$(the number of alternating permutations in $\mathfrak{S}_n$). alternating permutation is downup permutation.

Give a bijective proof-----the number of oriented increasing binary trees on the vertex set $\{$ $1,2,\cdots,n$ $\}$ is the Eulerian number $E_n$(the number of alternating permutations in $\mathfrak{S}_n$). alternating permutation is downup permutation.

Give a bijective proof-----the number of oriented increasing binary trees on the vertex set $\lbrace 1,2,\cdots,n\rbrace$ is the Eulerian number $E_n$(the number of alternating permutations in $\mathfrak{S}_n$). alternating permutation is downup permutation.

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ask for a simple bijective proof about oriented increasing binary trees

Give a bijective proof-----the number of oriented increasing binary trees on the vertex set $\{$ $1,2,\cdots,n$ $\}$ is the Eulerian number $E_n$(the number of alternating permutations in $\mathfrak{S}_n$). alternating permutation is downup permutation.