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+#+title: AMS review of 'A weighted interpretation for the super Catalan
+numbers' by Allen and Gheorghiciuc
+
+#+date: <2015-01-20>
+
+The super Catalan numbers are defined as $$ T(m,n) = {(2 m)! (2 n)!
+\over 2 m! n! (m + n)!}. $$
+
+   This paper has two main results. First a combinatorial interpretation
+of the super Catalan numbers is given: $$ T(m,n) = P(m,n) - N(m,n) $$
+where \(P(m,n)\) enumerates the number of 2-Motzkin paths whose \(m\)
+-th step begins at an even level (called \(m\)-positive paths) and
+\(N(m,n)\) those with \(m\)-th step beginning at an odd level
+(\(m\)-negative paths). The proof uses a recursive argument on the
+number of \(m\)-positive and -negative paths, based on a recursion of
+the super Catalan numbers appearing in [I. M. Gessel, J. Symbolic
+Comput. *14* (1992), no. 2-3, 179--194;
+[[http://www.ams.org/mathscinet/search/publdoc.html?r=1&pg1=MR&s1=1187230&loc=fromrevtext][MR1187230]]]:
+$$ 4T(m,n) = T(m+1, n) + T(m, n+1). $$ This result gives an expression
+for the super Catalan numbers in terms of numbers counting the so-called
+ballot paths. The latter sometimes are also referred to as the
+generalised Catalan numbers forming the entries of the Catalan triangle.
+
+   Based on the first result, the second result is a combinatorial
+interpretation of the super Catalan numbers \(T(2,n)\) in terms of
+counting certain Dyck paths. This is equivalent to a theorem, which
+represents \(T(2,n)\) as counting of certain pairs of Dyck paths, in [I.
+M. Gessel and G. Xin, J. Integer Seq. *8* (2005), no. 2, Article 05.2.3,
+13 pp.;
+[[http://www.ams.org/mathscinet/search/publdoc.html?r=1&pg1=MR&s1=2134162&loc=fromrevtext][MR2134162]]],
+and the equivalence is explained at the end of the paper by a bijection
+between the Dyck paths and the pairs of Dyck paths. The proof of the
+theorem itself is also done by constructing two bijections between Dyck
+paths satisfying certain conditions. All the three bijections are
+formulated by locating, removing and adding steps.
+
+Copyright notice: This review is published at
+http://www.ams.org/mathscinet-getitem?mr=3275875, its copyright owned by
+the AMS.