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Journal articles on the topic 'Catalan paths'

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Published: 4 June 2021
Last updated: 10 February 2022

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1

Woan, Wen-jin. "Area of Catalan paths." Discrete Mathematics 226, no. 1-3 (2001): 439–44. http://dx.doi.org/10.1016/s0012-365x(00)00162-x.

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2

Aval, J. C., and N. Bergeron. "Catalan paths and quasi-symmetric functions." Proceedings of the American Mathematical Society 131, no. 4 (2002): 1053–62. http://dx.doi.org/10.1090/s0002-9939-02-06634-0.

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3

Sturmfels, Bernd, Emmanuel Tsukerman, and Lauren Williams. "Symmetric matrices, Catalan paths, and correlations." Journal of Combinatorial Theory, Series A 144 (November 2016): 496–510. http://dx.doi.org/10.1016/j.jcta.2016.06.014.

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4

Deutsch, Emeric, and Marc Renault. "A Variation on t-Catalan Paths: 11071." American Mathematical Monthly 113, no. 5 (2006): 460. http://dx.doi.org/10.2307/27641962.

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5

Cheng, Szu-En, Sen-Peng Eu, and Tung-Shan Fu. "Area of Catalan paths on a checkerboard." European Journal of Combinatorics 28, no. 4 (2007): 1331–44. http://dx.doi.org/10.1016/j.ejc.2006.01.006.

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6

Bayer, Margaret, and Keith Brandt. "The Pill Problem, Lattice Paths, and Catalan Numbers." Mathematics Magazine 87, no. 5 (2014): 388–94. http://dx.doi.org/10.4169/math.mag.87.5.388.

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7

Chen, Zhi, and Hao Pan. "Identities involving weighted Catalan, Schröder and Motzkin paths." Advances in Applied Mathematics 86 (May 2017): 81–98. http://dx.doi.org/10.1016/j.aam.2016.11.011.

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8

Kung, Joseph P. S., and Anna de Mier. "Catalan lattice paths with rook, bishop and spider steps." Journal of Combinatorial Theory, Series A 120, no. 2 (2013): 379–89. http://dx.doi.org/10.1016/j.jcta.2012.08.010.

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9

MATSUMOTO, KENGO. "CUNTZ–KRIEGER ALGEBRAS AND A GENERALIZATION OF CATALAN NUMBERS." International Journal of Mathematics 24, no. 05 (2013): 1350040. http://dx.doi.org/10.1142/s0129167x13500407.

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For a directed graph G, we generalize the Catalan numbers by using the canonical generating partial isometries of the Cuntz–Krieger algebra [Formula: see text] for the transition matrix AGof the directed edges of G. The generalized Catalan numbers [Formula: see text] enumerate the number of Dyck paths for the graph G. Its generating functions will be studied.
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10

Dershowitz, Nachum, and Christian Rinderknecht. "The Average Height of Catalan Trees by Counting Lattice Paths." Mathematics Magazine 88, no. 3 (2015): 187–95. http://dx.doi.org/10.4169/math.mag.88.3.187.

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11

Díaz, Rafael, and Leonardo Cano. "Continous analogues for the binomial coefficients and the Catalan numbers." Applicable Analysis and Discrete Mathematics 13, no. 2 (2019): 542–68. http://dx.doi.org/10.2298/aadm161015019d.

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Using techniques from the theories of convex polytopes, lattice paths, and indirect influences on directed manifolds, we construct continuous analogues for the binomial coefficients and the Catalan numbers. Our approach for constructing these analogues can be applied to a wide variety of combinatorial sequences. As an application we develop a continuous analogue for the binomial distribution.
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12

Comelles, Josep M. "Forgotten paths: culture and ethnicity in Catalan mental health policies (1900-39)." History of Psychiatry 21, no. 4 (2010): 406–23. http://dx.doi.org/10.1177/0957154x09338083.

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13

Furdada, Glòria, and Joan Manuel Vilaplana. "Statistical prediction of maximum avalanche run-out distances from topographic data in the western Catalan Pyrenees (northeast Spain)." Annals of Glaciology 26 (1998): 285–88. http://dx.doi.org/10.3189/1998aog26-1-285-288.

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We present regression models to estimate maximum avalanche run-out distances in the western Catalan Pyrenees (northeast Spain). The regression equations have been calculated from topographic parameters of 216 well-known avalanche paths, mapped on avalanche location maps at a scale of 1: 50 000. Historical information about avalanche maximum run-out zones has been available for some avalanche paths hut most of them have been delimited by vegetation and geomorphological indices.The avalanche profiles, which are representative of avalanche paths, have been classified on the basis of avalanche run
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14

Furdada, Glòria, and Joan Manuel Vilaplana. "Statistical prediction of maximum avalanche run-out distances from topographic data in the western Catalan Pyrenees (northeast Spain)." Annals of Glaciology 26 (1998): 285–88. http://dx.doi.org/10.1017/s026030550001497x.

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We present regression models to estimate maximum avalanche run-out distances in the western Catalan Pyrenees (northeast Spain). The regression equations have been calculated from topographic parameters of 216 well-known avalanche paths, mapped on avalanche location maps at a scale of 1: 50 000. Historical information about avalanche maximum run-out zones has been available for some avalanche paths hut most of them have been delimited by vegetation and geomorphological indices.The avalanche profiles, which are representative of avalanche paths, have been classified on the basis of avalanche run
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15

Lehtonen, Erkko, and Tamás Waldhauser. "Associative spectra of graph algebras I." Journal of Algebraic Combinatorics 53, no. 3 (2021): 613–38. http://dx.doi.org/10.1007/s10801-020-01010-w.

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AbstractAssociative spectra of graph algebras are examined with the help of homomorphisms of DFS trees. Undirected graphs are classified according to the associative spectra of their graph algebras; there are only three distinct possibilities: constant 1, powers of 2, and Catalan numbers. Associative and antiassociative digraphs are described, and associative spectra are determined for certain families of digraphs, such as paths, cycles, and graphs on two vertices.
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16

BLANCO, R. "COMPLEXITY OF VILLAMAYOR'S ALGORITHM IN THE NON-EXCEPTIONAL MONOMIAL CASE." International Journal of Mathematics 20, no. 06 (2009): 659–78. http://dx.doi.org/10.1142/s0129167x09005480.

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We study monomial ideals, always locally given by a monomial, like a reasonable first step to estimate in general the number of monoidal transformations of Villamayor's algorithm of resolution of singularities. The resolution of a monomial ideal [Formula: see text] is interesting due to its identification with the particular toric problem [Formula: see text]. In the special case, when all the exponents ai are greater than or equal to the critical value c, we construct the largest branch of the resolution tree which provides an upper bound involving partial sums of Catalan numbers. This case wi
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17

Haglund, J., J. Morse, and M. Zabrocki. "A Compositional Shuffle Conjecture Specifying Touch Points of the Dyck Path." Canadian Journal of Mathematics 64, no. 4 (2012): 822–44. http://dx.doi.org/10.4153/cjm-2011-078-4.

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Abstract We introduce a q, t-enumeration of Dyck paths that are forced to touch the main diagonal at specific points and forbidden to touch elsewhere and conjecture that it describes the action of the Macdonald theory ∇ operator applied to a Hall–Littlewood polynomial. Our conjecture refines several earlier conjectures concerning the space of diagonal harmonics including the “shuffle conjecture” (Duke J. Math. 126 (2005), pp. 195 − 232) for ∇ en[X]. We bring to light that certain generalized Hall–Littlewood polynomials indexed by compositions are the building blocks for the algebraic combinato
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18

Soler, Josep. "The anonymity of Catalan and the authenticity of Estonian: two paths for the development of medium-sized languages." International Journal of Bilingual Education and Bilingualism 16, no. 2 (2013): 153–63. http://dx.doi.org/10.1080/13670050.2012.720665.

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19

Fagard, Benjamin. "É vida, olha…: Imperatives as discourse markers and grammaticalization paths in Romance." Languages in Contrast 10, no. 2 (2010): 245–67. http://dx.doi.org/10.1075/lic.10.2.07fag.

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In this paper, we investigate the evolution from imperatives to discourse markers in Romance, with a corpus-based approach. We focus on the case of items coming from verbs meaning ‘to look’, in a semasiological perspective: Spanish and Catalan mira, Portuguese olha, Italian guarda, French regarde, Romanian uite. We show that they all share many uses, among which turn-taking, introduction of reported speech, hesitation phenomenon, topic-shifting and modalization, except for French regarde. We then establish (against Waltereit, 2002) that the development of these uses is the result of a process
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20

Pons-Rigat, Aleix, Sergi Saurí, and Mateu Turró. "Matching Funding, Mobility, and Spatial Equity Objectives in a Networkwide Road Pricing Model." Transportation Research Record: Journal of the Transportation Research Board 2606, no. 1 (2017): 1–8. http://dx.doi.org/10.3141/2606-01.

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A review of the pricing and funding model for interurban roads in Catalonia, Spain, is timely because of the forthcoming finalization of toll concessions and the decrease in road taxes. This paper proposes a comprehensive policy approach for a new networkwide road pricing model to simultaneously deal with financial sustainability, mobility management, and spatial equity. A multiobjective perspective involving multiple trade-offs is required. The rationale for these trade-offs is analyzed and a diagnosis of the Catalan road network is performed on the basis of a revenues–costs matrix, which is
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21

Deutsch, Emeric, David Callan, Sabin Cautis, and Southwest Missouri Problems Group. "Another Path to Generalized Catalan Numbers: 10751." American Mathematical Monthly 108, no. 9 (2001): 872. http://dx.doi.org/10.2307/2695568.

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22

Sulanke, Robert A. "Catalan path statistics having the Narayana distribution." Discrete Mathematics 180, no. 1-3 (1998): 369–89. http://dx.doi.org/10.1016/s0012-365x(97)00126-x.

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23

Romagny, Adrien, Laurent Jolivet, Armel Menant, et al. "Detailed tectonic reconstructions of the Western Mediterranean region for the last 35 Ma, insights on driving mechanisms." BSGF - Earth Sciences Bulletin 191 (2020): 37. http://dx.doi.org/10.1051/bsgf/2020040.

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Slab retreat, slab tearing and interactions of slabs are first-order drivers of the deformation of the overriding lithosphere. An independent description of the tectonic evolution of the back-arc and peripheral regions is a pre-requisite to test the proposed conceptual, analogue and numerical models of these complex dynamics in 3-D. We propose here a new series of detailed kinematics and tectonic reconstructions from 35 Ma to the Present shedding light on the driving mechanisms of back-arc rifting in the Mediterranean where several back-arc basins all started to form in the Oligocene. The step
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24

Sulanke, Robert A. "Constraint-sensitive Catalan path statistics having the Narayana distribution." Discrete Mathematics 204, no. 1-3 (1999): 397–414. http://dx.doi.org/10.1016/s0012-365x(98)00382-3.

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25

Shablya, Yuriy, and Dmitry Kruchinin. "Euler–Catalan’s Number Triangle and Its Application." Symmetry 12, no. 4 (2020): 600. http://dx.doi.org/10.3390/sym12040600.

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In this paper, we study such combinatorial objects as labeled binary trees of size n with m ascents on the left branch and labeled Dyck n-paths with m ascents on return steps. For these combinatorial objects, we present the relation of the generated number triangle to Catalan’s and Euler’s triangles. On the basis of properties of Catalan’s and Euler’s triangles, we obtain an explicit formula that counts the total number of such combinatorial objects and a bivariate generating function. Combining the properties of these two number triangles allows us to obtain different combinatorial objects th
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26

Perperi, A. A., N. M. Yavorskaya, and P. V. Yavorskyy. "RESEARCH OF GEOMETRY IN THE ARCHITECTURE OF ANTONIO GAUDI." Problems of theory and history of architecture of Ukraine, no. 20 (May 12, 2020): 312–21. http://dx.doi.org/10.31650/2519-4208-2020-20-312-321.

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Exploring the work of Antonio Gaudi and his knowledge in descriptive geometry, which develops a spatial imagination, we notice the use of lines, shapes, volumetric geometric images in architectural creativity. In his early work, Antonio Gaudi designs buildings under the influence of the Moorish style, using geometric shapes and lines that emphasize Arabic motifs in the patterns. So in the house of Vissenty, made in the style of modernism, he applied the idea of a self-repeating element of a simple form toa more complex one. In a fragment of the gallery of the city park Guell, which is an examp
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27

Heller, Sarah-Grace. "Mocking Medieval French Fashion." French Historical Studies 43, no. 2 (2020): 145–66. http://dx.doi.org/10.1215/00161071-8018441.

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Abstract A debate poem (partimen PC 16,17) from the early thirteenth century richly demonstrates attitudes toward medieval French fashion, debating who was better, the “French” of France and England for their sumptuous apparel and generous feasts or the “Catalan” Occitan speakers for whom acquisition meant jovial pillage. Fashion appears as a preoccupation of the north, in contrast with southern poverty. Examined in context with political sirventes poems celebrating plunder, the Chanson de la croisade albigeoise (ca. 1210–12), which expresses pathos through clothing tropes, and the hyperbolic
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28

Santos, Jean Carlos Vieira, Vandervilson Alves Carneiro, Lorranne Gomes da Silva, and Vinícius Polzin Druciaki. "PAISAGENS E EMPIRISMO, ELEMENTOS CONSTITUTIVOS DAS EXPERIÊNCIAS EM GEOGRAFIA: um estudo sobre as veredas do chapadão de Catalão (Goiás)." Revista Cerrados 16, no. 02 (2020): 213–39. http://dx.doi.org/10.22238/rc2448269220181602213239.

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O presente artigo apresenta os resultados de um trabalho de campo que teve como área de estudo as veredas do chapadão de Catalão (Goiás, Brasil), nos municípios de Goiandira, Catalão e Ipameri, onde as mudanças no uso do solo são constantes, pois as chapadas são formas de relevo mecanizadas, adubadas e irrigadas que sofreram grande emprego de agrotóxicos nas últimas décadas. Partiu-se do levantamento bibliográfico, fotográfico e de observações empíricas, sendo que o desenvolvimento desta investigação vem ao encontro do método e das técnicas das pesquisas qualitativas que enfatizam a participaç
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29

Santos, Thallita Alves dos, Maria Tays Pereira Santana, Lucas Linhares Gomes, et al. "Avaliação da presença de Staphylococcus coagulase-positiva em diferentes superfícies do ambiente clínico odontológico." Research, Society and Development 10, no. 2 (2021): e47210212694. http://dx.doi.org/10.33448/rsd-v10i2.12694.

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O presente estudo tem como objetivo avaliar a presença de cepas de Staphylococcus coagulase-positiva (SCP) no ambiente de atendimento da Clínica Escola de Odontologia da Universidade Federal de Campina Grande – Campus Patos. Na metodologia, a coleta foi realizada com auxílio de swabs estéreis em diferentes superfícies do ambiente clínico odontológico e para identificação das cepas de SCP foram realizados os seguintes testes: coloração de Gram, fermentação do ágar manitol, produção de catalase e coagulase em lâmina. Obtivemos como resultado a presença de SCP em todos os boxes de atendimento odo
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30

Sopamena, Patma, Toto Nusantara, Eddy Bambang Irawan, and Sisworo. "Students’ thinking path in mathematics problem-solving referring to the construction of reflective abstraction." Beta: Jurnal Tadris Matematika 11, no. 2 (2018): 155–66. http://dx.doi.org/10.20414/betajtm.v11i2.230.

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[English]: This research aims to reveal the path of student thinking in solving mathematical problems referring to the construction of reflective abstraction. Reflective abstraction is the process of thinking in constructing logical structures (logico-mathematical structures) by individuals through interiorization, coordination, encapsulation, and generalization. It is an explorative research with the qualitative descriptive approach which involve fourteen undergraduate students enrolled in Calculus course. Data was analyzed through (1) transcribing verbal data (results of think aloud, intervi
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31

Bhatia, Kapil, Vivek N. Ambade, Alka Sontakke, and Dashrath Basannar. "Association between various antioxidants in normals and chronic obstructive pulmonary disease, their alteration and impact of smoking and disease on levels of antioxidants." International Journal of Research in Medical Sciences 7, no. 10 (2019): 3610. http://dx.doi.org/10.18203/2320-6012.ijrms20194288.

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Background: There is total alteration of various antioxidants in response to the oxidative stress, which is one of the major patho-physiologic hallmarks in chronic obstructive pulmonary disease (COPD) development. This study aims to establish the correlation between different antioxidants in normals and COPD, study the alteration in the correlation due to COPD and smoking as well as the impact of COPD and smoking on antioxidants levels.Methods: Study comprises of 96 normals as group I and 96 COPD patients as group II. The antioxidants albumin (Alb), bilirubin (Bil), uric acid (UA) ceruloplasmi
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32

Frederiks, W. M., and F. Marx. "A histochemical procedure for light microscopic demonstration of xanthine oxidase activity in unfixed cryostat sections using cerium ions and a semipermeable membrane technique." Journal of Histochemistry & Cytochemistry 41, no. 5 (1993): 667–70. http://dx.doi.org/10.1177/41.5.8468447.

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Xanthine oxidoreductase exists in two functionally distinct forms. Under normal conditions, the larger part of the enzyme occurs as an NAD(+)-dependent dehydrogenase form which produces NADH and urate. The dehydrogenase can be transformed under various (patho)physiological conditions to an oxygen-dependent oxidase form which produces oxygen radicals and/or hydrogen peroxide and urate. Tetrazolium salts are used to demonstrate the total activity of both the dehydrogenase and the oxidase form of the enzyme. We have developed a procedure to detect the oxidase form only in unfixed cryostat section
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33

Masudi, Idris. "Islam Dibawa Masuk oleh Orang Nusantara: Dari Data Terserak Buzurgh Al-Ramahurmuzi, ‘Ajaibul Hind: Kisah-Kisah Ajaib di Daratan dan Lautan Hindi." ISLAM NUSANTARA: Journal for Study of Islamic History and Culture 1, no. 1 (2020): 239–52. http://dx.doi.org/10.47776/islamnusantara.v1i1.52.

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Studies of the archipelago (nusantara) on the notes of foreign travelers written in the 9th and 10th of centuries are still quite rare. Indeed, there have been several studies on the notes of travelers such as Ma Huan (China), Tome Pires (Portuguese), Ibn Bathuthah (Arabic), and some others. But, these studies revolve around the notes of travelers after the 10th of century. Meanwhile, notes of travelers who came to the archipelago (nusantara) in the century before 10 AD have not got serious attention yet. This book is a travel note's report which captures various activities in India, China and
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34

Tsukerman, Emmanuel, Lauren Williams, and Bernd Sturmfels. "Symmetric matrices, Catalan paths, and correlations." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings, 28th... (April 22, 2020). http://dx.doi.org/10.46298/dmtcs.6337.

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International audience Kenyon and Pemantle (2014) gave a formula for the entries of a square matrix in terms of connected principal and almost-principal minors. Each entry is an explicit Laurent polynomial whose terms are the weights of domino tilings of a half Aztec diamond. They conjectured an analogue of this parametrization for symmetric matrices, where the Laurent monomials are indexed by Catalan paths. In this paper we prove the Kenyon-Pemantle conjecture, and apply this to a statistics problem pioneered by Joe (2006). Correlation matrices are represented by an explicit bijection from th
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35

Brignall, Robert, and Jakub Sliačan. "Juxtaposing Catalan Permutation Classes with Monotone Ones." Electronic Journal of Combinatorics 24, no. 2 (2017). http://dx.doi.org/10.37236/6625.

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This paper enumerates all juxtaposition classes of the form "$\mathrm{Av}(abc)$ next to $\mathrm{Av}(xy)$", where $abc$ is a permutation of length three and $xy$ is a permutation of length two. We use Dyck paths decorated by sequences of points to represent elements from such a juxtaposition class. Context free grammars are then used to enumerate these decorated Dyck paths.
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36

Bandlow, Jason, and Kendra Killpatrick. "An Area-to-Inv Bijection Between Dyck Paths and 312-avoiding Permutations." Electronic Journal of Combinatorics 8, no. 1 (2001). http://dx.doi.org/10.37236/1584.

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The symmetric $q,t$-Catalan polynomial $C_n(q,t)$, which specializes to the Catalan polynomial $C_n(q)$ when $t=1$, was defined by Garsia and Haiman in 1994. In 2000, Garsia and Haglund described statistics $a(\pi)$ and $b(\pi)$ on Dyck paths such that $C_n(q,t) = \sum_{\pi} q^{a(\pi)}t^{b(\pi)}$ where the sum is over all $n \times n$ Dyck paths. Specializing $t=1$ gives the Catalan polynomial $C_n(q)$ defined by Carlitz and Riordan and further studied by Carlitz. Specializing both $t=1$ and $q=1$ gives the usual Catalan number $C_n$. The Catalan number $C_n$ is known to count the number of $n
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37

Loehr, Nicholas, and Gregory Warrington. "Sweep maps for lattice paths." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings vol. AT,..., Proceedings (2014). http://dx.doi.org/10.46298/dmtcs.2432.

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International audience Sweep maps are a family of maps on words that, while simple to define, are not yet known to be injective in general. This family subsumes many of the "zeta maps" that have arisen in the study of q,t-Catalan numbers in the course of relating the three statistics of area, bounce and dinv. A sweep map can be defined for words over arbitrary alphabets with arbitrary weights. The latter property makes them particularly suitable for the study of rational Catalan combinatorics.
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38

Egge, E. S., J. Haglund, K. Killpatrick, and D. Kremer. "A Schröder Generalization of Haglund's Statistic on Catalan Paths." Electronic Journal of Combinatorics 10, no. 1 (2003). http://dx.doi.org/10.37236/1709.

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Garsia and Haiman (J. Algebraic. Combin. $\bf5$ $(1996)$, $191-244$) conjectured that a certain sum $C_n(q,t)$ of rational functions in $q,t$ reduces to a polynomial in $q,t$ with nonnegative integral coefficients. Haglund later discovered (Adv. Math., in press), and with Garsia proved (Proc. Nat. Acad. Sci. ${\bf98}$ $(2001)$, $4313-4316$) the refined conjecture $C_n(q,t) = \sum q^{{\rm area}}t^{{\rm bounce}}$. Here the sum is over all Catalan lattice paths and ${\rm area}$ and ${\rm bounce}$ have simple descriptions in terms of the path. In this article we give an extension of $({\rm area},{
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39

Song, Chunwei. "The Generalized Schröder Theory." Electronic Journal of Combinatorics 12, no. 1 (2005). http://dx.doi.org/10.37236/1950.

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While the standard Catalan and Schröder theories both have been extensively studied, people have only begun to investigate higher dimensional versions of the Catalan number (see, say, the 1991 paper of Hilton and Pedersen, and the 1996 paper of Garsia and Haiman). In this paper, we study a yet more general case, the higher dimensional Schröder theory. We define $m$-Schröder paths, find the number of such paths from $(0,0)$ to $(mn, n)$, and obtain some other results on the $m$-Schröder paths and $m$-Schröder words. Hoping to generalize classical $q$-analogue results of the ordinary Catalan and
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40

Callan, David. "A Bijection on Dyck Paths and its Cycle Structure." Electronic Journal of Combinatorics 14, no. 1 (2007). http://dx.doi.org/10.37236/946.

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The known bijections on Dyck paths are either involutions or have notoriously intractable cycle structure. Here we present a size-preserving bijection on Dyck paths whose cycle structure is amenable to complete analysis. In particular, each cycle has length a power of 2. A new manifestation of the Catalan numbers as labeled forests crops up en route as does the Pascal matrix mod 2. We use the bijection to show the equivalence of two known manifestations of the Motzkin numbers. Finally, we consider some statistics on the new Catalan manifestation and the identities they interpret.
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41

Xin, Guoce, and Jing-Feng Xu. "A Short Approach to Catalan Numbers Modulo $2^r$." Electronic Journal of Combinatorics 18, no. 1 (2011). http://dx.doi.org/10.37236/664.

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We notice that two combinatorial interpretations of the well-known Catalan numbers $C_n=(2n)!/n!(n+1)!$ naturally give rise to a recursion for $C_n$. This recursion is ideal for the study of the congruences of $C_n$ modulo $2^r$, which attracted a lot of interest recently. We present short proofs of some known results, and improve Liu and Yeh's recent classification of $C_n$ modulo $2^r$. The equivalence $C_{n}\equiv_{2^r} C_{\bar n}$ is further reduced to $C_{n}\equiv_{2^r} C_{\tilde{n}}$ for simpler $\tilde{n}$. Moreover, by using connections between weighted Dyck paths and Motzkin paths, we
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42

Pan, Qiongqiong, and Jiang Zeng. "On Total Positivity of Catalan-Stieltjes Matrices." Electronic Journal of Combinatorics 23, no. 4 (2016). http://dx.doi.org/10.37236/6270.

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Recently Chen-Liang-Wang (Linear Algerbra Appl. 471 (2015) 383—393) proved some sufficient conditions for the total positivity of Catalan-Stieltjes matrices. Our aim is to provide a combinatorial interpretation of their sufficiant conditions. More precisely, for any Catalan-Stieltjes matrix $A$ we construct a digraph with a weight, which is positive under their sufficient conditions, such that every minor of $A$ is equal to the sum of weights of families of nonintersecting paths of the digraph. We have also an analogue result for the minors of Hankel matrix associated to the first column of Ca
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43

Niederhausen, Heinrich. "Catalan Traffic at the Beach." Electronic Journal of Combinatorics 9, no. 1 (2002). http://dx.doi.org/10.37236/1649.

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We design a lattice path problem in ${\bf Z}^{2}$ (the Catalan traffic) with step set $\{\rightarrow,\uparrow\}$ strictly above the line $y=( x-1) /2$, and with step set $\{\downarrow,\searrow\}$ below that same line, except for the gates at $(2y,y)$ (with $\{\uparrow,\downarrow ,\searrow\}$-steps) and the closed intersections at $( 2y+1,y) $ (no traffic). The step sets prevent any traffic from going below the diagonal $y=-x$ (the beach). If we denote by $t(n,m)$ the number of paths from the origin to $(n,m)$, then the ubiquitous Catalan numbers $C_{n}={{2n}\choose {n}}/ ( n+1) $ occur as $t(
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44

Armstrong, Drew, Brendon Rhoades, and Nathan Williams. "Rational Catalan Combinatorics: The Associahedron." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings vol. AS,..., Proceedings (2013). http://dx.doi.org/10.46298/dmtcs.2355.

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International audience Each positive rational number $x>0$ can be written $\textbf{uniquely}$ as $x=a/(b-a)$ for coprime positive integers 0<$a$<$b$. We will identify $x$ with the pair $(a,b)$. In this extended abstract we use $\textit{rational Dyck paths}$ to define for each positive rational $x>0$ a simplicial complex $\mathsf{Ass} (x)=\mathsf{Ass} (a,b)$ called the $\textit{rational associahedron}$. It is a pure simplicial complex of dimension $a-2$, and its maximal faces are counted by the $\textit{rational Catalan number}$ $\mathsf{Cat} (x)=\mathsf{Cat}(a,b):=\frac{(a+b-1)! }{
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45

Lee, Kyungyong, Li Li, and Nicholas A. Loehr. "Limits of Modified Higher $q,t$-Catalan Numbers." Electronic Journal of Combinatorics 20, no. 3 (2013). http://dx.doi.org/10.37236/3201.

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The $q,t$-Catalan numbers can be defined using rational functions, geometry related to Hilbert schemes, symmetric functions, representation theory, Dyck paths, partition statistics, or Dyck words. After decades of intensive study, it was eventually proved that all these definitions are equivalent. In this paper, we study the similar situation for higher $q,t$-Catalan numbers, where the equivalence of the algebraic and combinatorial definitions is still conjectural. We compute the limits of several versions of the modified higher $q,t$-Catalan numbers and show that these limits equal the genera
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46

Gorsky, Eugene, Mikhail Mazin, and Monica Vazirani. "Rational Dyck Paths in the Non Relatively Prime Case." Electronic Journal of Combinatorics 24, no. 3 (2017). http://dx.doi.org/10.37236/6901.

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We study the relationship between rational slope Dyck paths and invariant subsets of $\mathbb{Z},$ extending the work of the first two authors in the relatively prime case. We also find a bijection between $(dn,dm)$–Dyck paths and $d$-tuples of $(n,m)$-Dyck paths endowed with certain gluing data. These are the first steps towards understanding the relationship between rational slope Catalan combinatorics and the geometry of affine Springer fibers and knot invariants in the non relatively prime case.
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47

Gorsky, Eugene, Mikhail Mazin, and Monica Vazirani. "Rational Dyck Paths in the Non Relatively Prime Case." Discrete Mathematics & Theoretical Computer Science DMTCS Proceedings, 28th... (April 22, 2020). http://dx.doi.org/10.46298/dmtcs.6387.

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International audience We study the relationship between rational slope Dyck paths and invariant subsets in Z, extending the work of the first two authors in the relatively prime case. We also find a bijection between (dn, dm)–Dyck paths and d-tuples of (n, m)-Dyck paths endowed with certain gluing data. These are first steps towards understanding the relationship between the rational slope Catalan combinatorics in non relatively prime case and the geometry of affine Springer fibers and representation theory.
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48

Brak, R., and J. W. Essam. "Bicoloured Dyck Paths and the Contact Polynomial for $n$ Non-Intersecting Paths in a Half-Plane Lattice." Electronic Journal of Combinatorics 10, no. 1 (2003). http://dx.doi.org/10.37236/1728.

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In this paper configurations of $n$ non-intersecting lattice paths which begin and end on the line $y=0$ and are excluded from the region below this line are considered. Such configurations are called Hankel $n-$paths and their contact polynomial is defined by $\hat{Z}^{\cal{H}}_{2r}(n;\kappa)\equiv \sum_{c= 1}^{r+1} |{\cal H}_{2r}^{(n)}(c)|\kappa^c$ where ${\cal H}_{2r}^{(n)}(c)$ is the set of Hankel $n$-paths which make $c$ intersections with the line $y=0$ the lowest of which has length $2r$. These configurations may also be described as parallel Dyck paths. It is found that replacing $\kap
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49

Chen, William Y. C., Toufik Mansour, and Sherry H. F. Yan. "Matchings Avoiding Partial Patterns." Electronic Journal of Combinatorics 13, no. 1 (2006). http://dx.doi.org/10.37236/1138.

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We show that matchings avoiding a certain partial pattern are counted by the $3$-Catalan numbers. We give a characterization of $12312$-avoiding matchings in terms of restrictions on the corresponding oscillating tableaux. We also find a bijection between matchings avoiding both patterns $12312$ and $121323$ and Schröder paths without peaks at level one, which are counted by the super-Catalan numbers or the little Schröder numbers. A refinement of the super-Catalan numbers is derived by fixing the number of crossings in the matchings. In the sense of Wilf-equivalence, we use the method of gene
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50

Alexandersson, Per, Svante Linusson, and Samu Potka. "The Cyclic Sieving Phenomenon on Circular Dyck Paths." Electronic Journal of Combinatorics 26, no. 4 (2019). http://dx.doi.org/10.37236/8720.

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We give a $q$-enumeration of circular Dyck paths, which is a superset of the classical Dyck paths enumerated by the Catalan numbers. These objects have recently been studied by Alexandersson and Panova. Furthermore, we show that this $q$-analogue exhibits the cyclic sieving phenomenon under a natural action of the cyclic group. The enumeration and cyclic sieving is generalized to Möbius paths. We also discuss properties of a generalization of cyclic sieving, which we call subset cyclic sieving, and introduce the notion of Lyndon-like cyclic sieving that concerns special recursive properties of
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