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The first few eulerian catalan numbers, beginning with

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The first few eulerian catalan numbers, beginning with EC_0=1, are 1,2,22,604,31238. Show that EC_n=2A(2n,n+1), where Ec_n is any integers. The eulerian catalan number is ***** by EC_n=A(2n+1, n+1)/(n+1). Also show that EC_n is the number of permutations w=a_1a_2...a_{2n+1} with n descents, such that every left factor a1a2...ai has at least as many ascents as descents. For n=1 we are counting the two permutations 132 and 231.

I can only refer you to published (or semi-published, such as arXiv) papers. I can't really explain the proof in my own words. As per our code of conduct, I cannot take credit for that, so I'll have to decline to answer.