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Enumerative Combinatorics by Sergey Fomin, Richard P. Stanley

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6.34. There are many possible q-analogues of Catalan numbers. In (a) we give what is perhaps the most natural “combinatorial” q-analogue, while in (b) we give the most natural “explicit formula” q-analogue. In (c) we give an interesting extension of (b), while (d) and (e) are concerned with another special case of (c).
a. [2+] Let

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where the sum is over all lattice paths P from (0, 0) to (n, n) with steps (1, 0) and (0, 1), such that P never rises above the line y = x, and where A(P) is the area under the path (and above the x-axis). Note that by Exercise 6.19(h), we have Cn(l) = Cn. (It is interesting to see what statistic corresponds to A(

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