8,381 research outputs found
Growing and destroying Catalan–Stanley trees
No full textCITATION: Hack, B. & Prodinger, H. 2018. Growing and destroying catalan–stanley trees. Discrete Mathematics and Theoretical Computer Science, 20(1):1-14, doi:10.23638/DMTCS-20-1-11.The original publication is available at https://www.semanticscholar.orgStanley lists the class of Dyck paths where all returns to the axis are of odd length as one of the many objects enumerated
by (shifted) Catalan numbers. By the standard bijection in this context, these special Dyck paths correspond
to a class of rooted plane trees, so-called Catalan–Stanley trees.
This paper investigates a deterministic growth procedure for these trees by which any Catalan–Stanley tree can be
grown from the tree of size one after some number of rounds; a parameter that will be referred to as the age of the
tree. Asymptotic analyses are carried out for the age of a random Catalan–Stanley tree of given size as well as for the
“speed” of the growth process by comparing the size of a given tree to the size of its ancestors.https://www.semanticscholar.org/paper/Growing-and-Destroying-Catalan-Stanley-Trees-Hackl-Prodinger/eced0779211103ebf2752dc775e2a91b5beb5d73Publisher's versio
Down-step statistics in generalized Dyck paths
No full textThe number of down-steps between pairs of up-steps in -Dyck paths, a
generalization of Dyck paths consisting of steps such
that the path stays (weakly) above the line , is studied. Results are
proved bijectively and by means of generating functions, and lead to several
interesting identities as well as links to other combinatorial structures. In
particular, there is a connection between -Dyck paths and perforation
patterns for punctured convolutional codes (binary matrices) used in coding
theory. Surprisingly, upon restriction to usual Dyck paths this yields a new
combinatorial interpretation of Catalan numbers
Binomial Sums and Mellin Asymptotics with Explicit Error Bounds: A Case Study
No full textMaking use of a newly developed package in the computer algebra system SageMath, we show how to perform a full asymptotic analysis by means of the Mellin transform with explicit error bounds. As an application of the method, we answer a question of Bóna and DeJonge on 132-avoiding permutations with a unique longest increasing subsequence that can be translated into an inequality for a certain binomial sum
J.C. Painter letter to Benjamin Lundy
No full textLetter from J.E. Painter to (presumably) Benjamin Lundy, answering a request for information about the history and operations of the Underground Railroad. Letter includes details of a story of an ex-slave transported on the Underground Railroad through Ohio and stories of the plight of other fugitive slaves crossing the Ohio River.
Benjamin Lundy (1789-1839) was a prominent Quaker abolitionist best known for his development of abolitionist periodicals. His "Genius of Universal Emancipation" was first published in 1821 from his home in Mt. Pleasant, Ohio, and enjoyed a wide circulation across the antebellum United States. In the 1820s, the young William Lloyd Garrison came to work for The Genius. Benjamin Lundy traveled widely seeking subscriptions to The Genius, giving talks about the anti-slavery movement, and observing and documenting the conditions of enslaved people across the Americas. He was also involved in the establishment of freed slave colonies in Mexico
Mexican land grant contract to Benjamin Lundy, March 10, 1835 (English)
No full textLegal document from an unsigned officer to Benjamin Lundy, authorizing him rights as empresario to a tract of land in then-Mexico. The document extends a previous treaty made to Lundy by the government of Mexico from November 17, 1823 -- presumably, this land is to be the site of Lundy's freed slave colony. Original Spanish-language document is also a part of this collection. Benjamin Lundy (1789-1839) was a prominent Quaker abolitionist best known for his development of abolitionist periodicals. His Genius of Universal Emancipation was first published in 1821 from his home in Mt. Pleasant, Ohio, and enjoyed a wide circulation across the antebellum United States. In the 1820s, the young William Lloyd Garrison came to work for The Genius. Benjamin Lundy traveled widely seeking subscriptions to The Genius, giving talks about the anti-slavery movement, and observing and documenting the conditions of enslaved people across the Americas. He was also involved in the establishment of freed slave colonies in Mexico
Eli Nichols letter to Benjamin Lundy, March 17th, 1839
No full textFriendly note from Eli Nichols to Benjamin Lundy covering topics in contemporary abolition, ranging from the social status of abolitionists to the oppression of the poor. Much of the letter concerns a review of contemporary social movements in equality-based education, including Shaker and Quaker communities. The letter concludes in discussion of Nichols' and Lundy's interest in forming a freed slave colony or community in then-Mexico, and describes the climate and culture of those regions in detail. Benjamin Lundy (1789-1839) was a prominent Quaker abolitionist best known for his development of abolitionist periodicals. His Genius of Universal Emancipation was first published in 1821 from his home in Mt. Pleasant, Ohio, and enjoyed a wide circulation across the antebellum United States. In the 1820s, the young William Lloyd Garrison came to work for The Genius. Benjamin Lundy traveled widely seeking subscriptions to The Genius, giving talks about the anti-slavery movement, and observing and documenting the conditions of enslaved people across the Americas. He was also involved in the establishment of freed slave colonies in Mexico
Mexican land grant contract to Benjamin Lundy, March 10, 1835 (Spanish)
No full textLegal document in Spanish from the government of Tamaulipas, Mexico, to Benjamin Lundy, which appears to grant Lundy the rights of empresario for his proposed colony for freed slaves in Tamaulipas. This document appears to be truncated; it ends abruptly after 2 pages. Collection also includes a period translation of this contract with Lundy in English, which appears to contain the full text of the agreement. Benjamin Lundy (1789-1839) was a prominent Quaker abolitionist best known for his development of abolitionist periodicals. His Genius of Universal Emancipation was first published in 1821 from his home in Mt. Pleasant, Ohio, and enjoyed a wide circulation across the antebellum United States. In the 1820s, the young William Lloyd Garrison came to work for The Genius. Benjamin Lundy traveled widely seeking subscriptions to The Genius, giving talks about the anti-slavery movement, and observing and documenting the conditions of enslaved people across the Americas. He was also involved in the establishment of freed slave colonies in Mexico
Uncovering a Random Tree
No full textWe consider the process of uncovering the vertices of a random labeled tree according to their labels. First, a labeled tree with n vertices is generated uniformly at random. Thereafter, the vertices are uncovered one by one, in order of their labels. With each new vertex, all edges to previously uncovered vertices are uncovered as well. In this way, one obtains a growing sequence of forests. Three particular aspects of this process are studied in this extended abstract: first the number of edges, which we prove to converge to a stochastic process akin to a Brownian bridge after appropriate rescaling. Second, the connected component of a fixed vertex, for which different phases are identified and limiting distributions determined in each phase. Lastly, the largest connected component, for which we also observe a phase transition
General Benjamin Butler Letter Regarding the naming of Newport News, Virginia
No full textDigital images of an original letter written by Former Union Major-General Benjamin Butler in reply to a query by author, Edwin Everett Hale on how Newport News, Virginia had received it's name. both sides of the original letter are included along with a typed transcription of the letter
Counting Ascents in Generalized Dyck Paths
No full textNon-negative Lukasiewicz paths are special two-dimensional lattice paths never passing below their starting altitude which have only one single special type of down step. They are well-known and -studied combinatorial objects, in particular due to their bijective relation to trees with given node degrees.
We study the asymptotic behavior of the number of ascents (i.e., the number of maximal sequences of consecutive up steps) of given length for classical subfamilies of general non-negative Lukasiewicz paths: those with arbitrary ending altitude, those ending on their starting altitude, and a variation thereof. Our results include precise asymptotic expansions for the expected number of such ascents as well as for the corresponding variance
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