WebThe Catalan numbers are a sequence of positive integers that appear in many counting problems in combinatorics. They count certain types of lattice paths, permutations, binary trees, and many other combinatorial objects. They satisfy a fundamental recurrence relation, and have a closed-form formula in terms of binomial coefficients. WebThe q, t-Catalan numbers arise naturally in the study of diagonal harmonics. They are conjectured to be the coefficient of the sign representation in the Frobenius series for the diagonal harmonics. The (q,t)-Catalan numbers are now known to be polynomials in q and t and there exists a proof in the article
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WebNov 19, 2024 · Abstract. In the paper, the author generalizes several formulas and series identities involving the Catalan numbers and establishes several new formulas and series identities involving the Catalan-Qi numbers and q -Catalan-Qi numbers. Download to read the full article text. WebMay 5, 2024 · I would like to know if the following recurrence relation for Catalan numbers (see mathoverflow.net/questions/191524 and also math.stackexchange.com/questions/2113830) has appeared in a paper or a book, so that I can cite it. C n = 1 + ∑ k = 1 ⌈ n 2 ⌉ ( − 1) k + 1 ( n − k k) C n − k where C n is the n -th … rothammel antenna book
enumerative meaning of natural q-Catalan numbers
WebJul 12, 2024 · Definition: Catalan Number The n th Catalan number, C n, is the number of different ways in which brackets can be put around n terms, to indicate different orders of combining the terms. Thus, for example, C 3 = 2, since three terms can be combined as either [ (_·_) ·_], or [_·_ (_·_)]. WebMay 16, 2024 · Clearly \lim _ {q\rightarrow 1}C_n (q)=C_n. C_n (q) is often called the Carlitz–Riordan q -Catalan number. The Carlitz–Riordan q -Catalan numbers also have … WebJan 6, 2024 · Hankel determinants for q-Catalan numbers where q is a root of unity? Ask Question Asked 1 year, 3 months ago. Modified 1 year, 3 months ago. Viewed 241 times 5 $\begingroup$ Let ${C_n}(q)$ be the weight ... st patty\u0027s day or st paddy\u0027s day