Journal of Integer Sequences, Vol. 4 (2001),Dyck Paths With No Peaks At Height k Paul Peart and Wen-Jin Woan Department of Mathematics Howard University Washington,D.C.20059,USA Email addresses:pp@scs.howard.edu ,wwoan@howard.edu Abstract A Dyck path of length 2n is a path in two-...
For any integer $k$ with $1 \\leq k \\leq n$, let $p_{n,m,k}$ be the total number of $(n,m)$-Dyck paths with $k$ peaks. Ma and Yeh proved that $p_{n,m,k}$=$p_{n,n-m,n-k}$ for $0 \\leq m \\leq n$, and $p_{n,m,k}+p_{n,m,n-k}=p_{n,m+1...
幸上1≤n—l 肚(Xy) (麓荛) n—k obtaina valleys wecan comments Q∈巩with Fromtheabore Dyckpath unique the we wefinda between complete toV+.So bijection corresponding 口 the proof. Table1. value illustratedin Thesmall of%^is value Table1:The ofⅣn≈ §2 Pathswith kPeaks 5 Dyck ...
(1�x).Itiseasytoseethatforanyk,Rk(x)isarationalfuntioninx.ADykpathisalattiepathintheplaneintegerlattieZ�Zonsistingofup-steps(1;1)anddown-steps(1;�1),whihneverpassesbelowthex-axis(seeFigure1).LetPbeaDy k path; w e de�ne the weight of P to b e the pro du t of the w ...
dyckrestrictiondyckpathpathspathswithdyckpaths 摘要 在这篇论文中我们集中讨论了有限制的Dyck路,推导出它们所满足的递推关系 式,生成函数,以及它们的公式。首先,我们研究了Narayana数,从Chu—Vandermonde 恒等式入手,得出了它的行生成函数与列生成函数之间的关系,由算子方法得出了一个 重要的关于Narayana数和Catalan数...
For integers n,m n , m mathContainer Loading Mathjax with n≥1 n ≥ 1 mathContainer Loading Mathjax and 0≤m≤n 0 ≤ m ≤ n mathContainer Loading Mathjax , an (n,m) ( n , m ) mathContainer Loading Mathjax -Dyck path is a lattice path in the integer lattice Z×Z Z × Z...
andSchroderpathsaleveryimportantcombinatorialstruc—turesinenumerativecombinatorics.Besidestheircloseconnectionswithothercombina-torialstructuresliketrees,restrictedpermutations,orthogonalpolynomials,continuedfrac-tions,etc.,theirapplicationsalsoarisefromfieldsasstatistics,randomprocessesandbioin—formatics.Inthispaperwemainly...
Inthispaperwestudytheenumerationofk-coloredskewDyckpaths.Wefirstgiveacountingformulaandarecurrenceforsandthelimitofs%/s一1.Wethengivethecountingformulasofthek-coloredskewDyckpathswithsemilengthnaccordingtothenumberofleftstepsandthenumberofpeaks,valleysanddoublerises.Keywords:Dyckpath;k-coloredskewDyckpath;...