Triangular array: Difference between revisions

Revision as of 21:14, 22 June 2024

Revision as of 12:36, 29 December 2024

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*The [[Bell triangle]], whose numbers count the [[Partition of a set|partitions of a set]] in which a given element is the largest [[singleton (mathematics)|singleton]]<ref>{{citation

| last = Shallit | first = Jeffrey | authorlink = Jeffrey Shallit

| editor1-first = Verner E. Jr. | editor1-last = Hoggatt

| editor2-first = Marjorie | editor2-last = Bicknell-Johnson

| contribution = A triangle for the Bell numbers

| ___location = Santa Clara, Calif.

Line 14 ⟶ 16:

| publisher = Fibonacci Association

| title = A collection of manuscripts related to the Fibonacci sequence

| url = httphttps://www.fq.math.ca/Books/Collection/shallitcollection.pdfhtml

| contribution-url = http://www.fq.math.ca/Books/Collection/shallit.pdf

| year = 1980}}.</ref>

* [[Catalan's triangle]], which counts strings of matched parentheses in which no close parenthesis is unmatched<ref>{{citation

| title = Harmonic numbers, Catalan's triangle and mesh patterns

| last1 = Kitaev | first1 = Sergey

| last1 = Kitaev | first1 = Sergey | author1-link = Sergey Kitaev

| last2 = Liese | first2 = Jeffrey

| doi = 10.1016/j.disc.2013.03.017

| issue = 14

| journal = [[Discrete Mathematics (journal)|Discrete Mathematics]]

| mryear = 30473902013

| pages = 1515–1531

| title = Harmonic numbers, Catalan's triangle and mesh patterns

| volume = 313

| issue = 14

| year = 2013| arxiv = 1209.6423

| pages = 1515–1531

| doi = 10.1016/j.disc.2013.03.017

| mr = 13637073047390

| year = 2013| arxiv = 1209.6423

| s2cid = 18248485

| url = https://personal.strath.ac.uk/sergey.kitaev/Papers/mesh1.pdf

}}.</ref>

* [[Euler's triangle]], which counts permutations with a given number of ascents<ref>{{citation

| title = Permutations and combination locks

| last1 = Velleman | first1 = Daniel J.

| last2 = Call | first2 = Gregory S.

| doi = 10.2307/2690567

| issue = 4

| journal = Mathematics Magazine

| year = 1995 | volume = 68 | issue = 4 | pages = 243–253

| mr = 1363707

| doi = 10.1080/0025570X.1995.11996328

| pages = 243–253

| yearmr = 19951363707 | jstor = 2690567

| title = Permutations and combination locks

}}.</ref>

| volume = 68

* [[Floyd's triangle]], whose entries are all of the integers in order<ref>{{citation

| year = 1995| jstor = 2690567

| title = Programming by design: a first course in structured programming

| pages = 1–34211–212

| first1=Philip L. | last1=Miller

| first2 = Lee W. | last2 = Miller

| first3 = Purvis M. | last3=Jackson

| publisher = Wadsworth Pub. Co.

| year = 1987

| isbn = 978-0-534-08244-4

}}.</ref>

* [[Floyd's triangle]], whose entries are all of the integers in order<ref>{{citation|title=Programming by design: a first course in structured programming|pages=211–212|first1=Philip L.|last1=Miller|first2=Lee W.|last2=Miller|first3=Purvis M.|last3=Jackson|publisher=Wadsworth Pub. Co.|year=1987|isbn=9780534082444}}.</ref>

* [[Hosoya's triangle]], based on the [[Fibonacci number]]s<ref>{{citation

| title = Fibonacci triangle

| last = Hosoya | first = Haruo | author-link = Haruo Hosoya

| issue = 2

| journal = [[The Fibonacci Quarterly]]

| pages = 173–178

| title = Fibonacci triangle

| volume = 14

| issue = 42

| pages = 243–253173–178

| year = 1976}}.</ref>

* [[Lozanić's triangle]], used in the mathematics of chemical compounds<ref>{{citation

| last = Losanitsch | first = S. M.

| journal = Chem. Ber.

| pages = 1917–1926

| title = Die Isomerie-Arten bei den Homologen der Paraffin-Reihe

| trans-title = The isomery species of the homologues of the paraffin series

| last = Losanitsch | first = Sima M. | author-link = Sima Lozanić

| journal = [[Chem. Ber.]] | lang = de

| volume = 30

| issue = 2

| year = 1897

| pages = 173–1781917–1926

| doi=10.1002/cber.189703002144| url = https://zenodo.org/record/1425862

| doi = 10.1002/cber.189703002144

| doi=10.1002/cber.189703002144| url = https://zenodo.org/record/1425862

}}.</ref>

* [[Narayana triangle]], counting strings of balanced parentheses with a given number of distinct nestings<ref>{{citation

| title = On a generalization of the Narayana triangle

| last = Barry | first = Paul

| journal = Journal of Integer Sequences

| issue = 4

| journal = Journal of Integer Sequences

| mr = 2792161

| page = Article 11.4.5, 22

| title = On a generalization of the Narayana triangle

| volume = 14

| article-number = 11.4.5

| mr = 2792161

| url = https://cs.uwaterloo.ca/journals/JIS/VOL14/Barry4/barry142.pdf

| year = 2011}}.</ref>

* [[Pascal's triangle]], whose entries are the [[binomial coefficients]]<ref>{{citation
| title = Pascal's Arithmetical Triangle: The Story of a Mathematical Idea
| first = A. W. F. | last = Edwards | author-link = A. W. F. Edwards
| publisher=JHU Press
| year=2002
| isbn =9780801869464 978-0-8018-6946-4}}.</ref>

Triangular arrays of integers in which each row is symmetric and begins and ends with 1 are sometimes called '''generalized Pascal triangles'''; examples include Pascal's triangle, the Narayana numbers, and the triangle of Eulerian numbers.<ref>{{citation

| last = Barry | first = P.

| issue = 6.2.4

| journal = Journal of Integer Sequences

| pages = 1–34

| title = On integer-sequence-based constructions of generalized Pascal triangles

| last = LosanitschBarry | first = S. M.Paul

| journal = Journal of Integer Sequences

| volume = 689 | issue = 2

| issuearticle-number = 6.2.4

| url = http://www.emis.de/journals/JIS/VOL9/Barry/barry91.pdf

| year = 2006 | bibcode = 2006JIntS...9...24B

| volume = 9

| year = 2006| bibcode = 2006JIntS...9...24B

}}.</ref>