Editing Catalan number (section)

== Generalizations ==
The Catalan numbers can be interpreted as a special case of the [[Bertrand's ballot theorem]]. Specifically, <math>C_n</math> is the number of ways for a candidate A with {{math|''n'' + 1}} votes to lead candidate B with {{mvar|n}} votes.

The two-parameter sequence of non-negative integers <math>\frac{(2m)!(2n)!}{(m+n)!m!n!}</math>
is a generalization of the Catalan numbers. These are named '''super-Catalan numbers''', per [[Ira Gessel]]. These should not  confused with the [[Schröder–Hipparchus number]]s, which sometimes are also called super-Catalan numbers.

For <math>m=1</math>, this is just two times the ordinary Catalan numbers, and for <math>m=n</math>, the numbers have an easy combinatorial description.
However, other combinatorial descriptions are only known<ref name="Chen2012">{{cite arXiv|last1=Chen|first1=Xin|last2=Wang|first2=Jane|title=The super Catalan numbers S(m, m + s) for s ≤ 4|year=2012|class=math.CO|eprint=1208.4196}}</ref>
for <math>m=2, 3</math> and <math>4</math>,<ref>{{cite arXiv|eprint=2008.00133|last1=Gheorghiciuc|first1=Irina|last2=Orelowitz|first2=Gidon|title=Super-Catalan Numbers of the Third and Fourth Kind|year=2020|class=math.CO}}</ref>
and it is an open problem to find a general combinatorial interpretation.

[[Sergey Fomin]] and Nathan Reading have given a generalized Catalan number associated to any finite crystallographic [[Coxeter group]], namely the number of fully commutative elements of the group; in terms of the associated [[root system]], it is the number of anti-chains (or order ideals) in the poset of positive roots. The classical Catalan number <math>C_n</math> corresponds to the root system of type <math>A_n</math>. The classical recurrence relation generalizes: the Catalan number of a Coxeter diagram is equal to the sum of the Catalan numbers of all its maximal proper sub-diagrams.<ref>[[Sergey Fomin]] and Nathan Reading, "Root systems and generalized associahedra", Geometric combinatorics, IAS/Park City Math. Ser. '''13''', [[American Mathematical Society]], Providence, RI, 2007, pp 63–131. {{arxiv|math/0505518}}</ref>

The Catalan numbers are a solution of a version of the [[Hausdorff moment problem]].<ref>{{citation
 | last1 = Choi | first1 = Hayoung
 | last2 = Yeh | first2 = Yeong-Nan
 | last3 = Yoo | first3 = Seonguk
 | arxiv = 1809.07523
 | doi = 10.1016/j.disc.2019.111808
 | issue = 5
 | journal = Discrete Mathematics
 | mr = 4052255
 | pages = 111808, 11
 | title = Catalan-like number sequences and Hausdorff moment sequences
 | volume = 343
 | year = 2020| s2cid = 214165563
 }}</ref>