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diff --git a/posts/2015-01-20-weighted-interpretation-super-catalan-numbers.md b/posts/2015-01-20-weighted-interpretation-super-catalan-numbers.md deleted file mode 100644 index 6d9e75e..0000000 --- a/posts/2015-01-20-weighted-interpretation-super-catalan-numbers.md +++ /dev/null @@ -1,40 +0,0 @@ ---- -template: oldpost -title: AMS review of 'A weighted interpretation for the super Catalan numbers' by Allen and Gheorghiciuc -date: 2015-01-20 -comments: true -archive: false ---- -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 \[I. M. Gessel, J. Symbolic Comput. **14** (1992), no. 2-3, 179–194; -[MR1187230](http://www.ams.org/mathscinet/search/publdoc.html?r=1&pg1=MR&s1=1187230&loc=fromrevtext)\]: -\$\$ 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 \[I. M. Gessel and G. -Xin, J. Integer Seq. **8** (2005), no. 2, Article -05.2.3, 13 pp.; -[MR2134162](http://www.ams.org/mathscinet/search/publdoc.html?r=1&pg1=MR&s1=2134162&loc=fromrevtext)\], -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. |