# Tagged Questions

The Catalan numbers form the sequence of numbers starting 1,1,2,5,14,42,... with explicit formula $\frac{1}{n+1}\binom{2n}{n}$. It counts many combinatorial objects like planar binary trees, triangulations, noncrossing partitions, Dyck paths, etc. See http://oeis.org/A000108

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### Solving for f given constraint involving f(x, y) and f(xy, y)

I am interested in a weighted version of the Catalan numbers. The generating function for this case, $$f(x, y) = \sum_s \sum_n f_{s n} x^s y^n$$ (where the $y^n$ term is the weight), obeys the ...
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### Is there a distance function on Dyck/Tamari words of arbitrary length?

Consider sequences of well-formed parentheses (or up/down sequences) of the type counted by the Catalan numbers. See http://www-math.mit.edu/~rstan/ec/catalan.pdf These are sometimes called Dyck ...
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### q-Catalan numbers from Grassmannians

In this question by $q$-Catalan numbers I mean the $q$-analog given by the formula $\frac{1}{[n+1]_q}\left[{2n\atop n}\right]_q$. The polynomial $\left[{2n\atop n}\right]_q$ represents the class of ...
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### Catalan numbers as sums of squares of numbers in the rows of the Catalan triangle - is there a combinatorial explanation?

This question arose from an answer to my recent question How many traces are there on Temperley-Lieb, Fuss-Catalan, Iwahori-Hecke, Birman-Wenzl-Murakami-Kauffman, ... algebras? What I need from that ...
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### Probability of a graph procedure

We are going to build $K_n$ one edge at a time. Begin with the empty graph on $n$ vertices. Take a random permutation of the edges of $K_n$ and, one at a time, place the edges onto the graph (so, ...
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### Lots of combinatorial interpretations of Catalan numbers

During a lecture I gave on Catalan numbers, I pointed out that that it is possible to give a continuum number of combinatorial interpretations of these numbers. See the solution to (f$^5$) on page 54 ...
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### When does a Catalan number equal a Fibonacci number?

The $n=3$'rd Catalan number (A000108) is $1,1,2,5$ : $\frac{\binom{2n}{n}}{n+1}=\frac{\binom{6}{3}}{4}=\frac{20}{4}=$ 5. The $n=4$'th Fibonacci number (A000045) is $1,1,2,3,5,...$ : 5. Q. Which ...
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### Number of Dyck paths with k returns and b peaks

The number of Dyck paths from the origin to $(2n,0)$ which touch the $x$-axis $k+2$ times ($k$ internal touches) is given by $$\frac{k}{2n-k}{2n-k \choose n}.$$ The number of Dyck paths from the ...
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### Non-abelian freeness of SU_2

The distribution of the trace of a random element of $SU_2$ is the Sato-Tate distribution. The analogue of the Gaussian distribution in free probability theory is the Wigner semicircle distribution. ...
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### More 3-connected cubic graphs with all 2-factors of same cycle type?

The setup is as in this question: Let $G$ be a 3-connected cubic graph. If all 2-factors of $G$ are isomorphic (as graphs), i.e. all have the same partition $\pi$ as cycle type, we'll say that $G$ is ...