cover

en-noun

cover (countable and uncountable, plural covers)

→ Hijazi Arabic: كَڤَر (kavar)

→ Portuguese: cover

Etymology tree
Proto-Indo-European *ḱe-?
Proto-Indo-European *ḱóm
Proto-Italic *kom
Old Latin com
Latin cum
Latin con-
Proto-Indo-European *h₁ep-?
Proto-Indo-European *h₁epi
Proto-Indo-European *h₂wer-
Proto-Indo-European *-yeti
Latin operiō
Latin cooperiō
Old French covrirbor.
Middle English coveren
English cover
From Middle English coveren, borrowed from Old French covrir, cueuvrir (modern French couvrir), from Late Latin coperire, from Latin cooperiō (“I cover completely”), from co- (intensive prefix) + operiō (“I close, cover”). Displaced native Middle English thecchen and bethecchen (“to cover”) (from Old English þeccan, beþeccan (“to cover”)), Middle English helen, (over)helen, (for)helen (“to cover, conceal”) (from Old English helan (“to conceal, cover, hide”)), Middle English wrien, (be)wreon (“to cover”) (from Old English (be)wrēon (“to cover”)), Middle English hodren, hothren (“to cover up”) (from Low German hudren (“to cover up”)).
According to the Oxford English Dictionary, the original sense of the verb and noun cover was “hide from view” as in its cognate covert. Except in the limited sense of “cover again”, the word recover is unrelated and is cognate with recuperate. Cognate with Spanish cubrir (“to cover”).

cumhdach

A cloth or similar material used to protect underlying items from dust, sunlight, insects etc. when not being used.

throw nickels around like manhole covers

The open intervals are a cover for the real numbers.

(combinatorics, topology) A collection (or family) of subsets of a given set, whose union contains every element of said original set.


permutation

en-noun

permutation (countable and uncountable, plural permutations)

/ˌpɜː.mjʉˈtæɪ.ʃən/

[ˌpɜː.mjʉˈtæɪ.ʃn̩]

From Middle English permutacioun, permutacyoun, from Old French permutacïon, promutatïon and Medieval Latin permūtātiōnem, accusative of permūtātiō.
Morphologically permute + -ation

There are six permutations of three elements, e.g. {abc, acb, bac, bca, cab, cba}.

(mathematics, combinatorics) An ordering of a finite set of distinct elements.


factorial

The expression #123;#125;#95;nC#95;r means the number of combinations of n things taken r at a time. It is also written #92;binomnr, and is equal to#92;frac#123;n(n-1)(n-2)(n-3)#92;cdots#92;cdots(n-r#43;1)#125;#123;r#33;#125;or n#33;#47;r#33;(n-r)#33;, in which r#33; is read "factorial r" and denotes the product of all the integral numbers from 1 to r inclusive.

1916 July, M. Mott-Smith, “The Arithmetical Pyramid of Many Dimensions”, in The Monist, volume 26, number 3, →DOI, page 428:

This playing around with consecutive integers reminded one of the students of factorials, and he asked about products of integers, which I said could be expressed similarly as#92;prod#95;#123;k#61;1#125;ⁿk#61;1#92;cdot 2#92;cdot 3#92;cdotsn.But what does this equal?
Well, of course we can say that it is equal to n factorial and write#92;prod#95;#123;k#61;1#125;ⁿk#61;n#33; and then use our calculators, but inventing a new symbol like 'factorial' feels like an admission of defeat!

2018 September, Colin Foster, “What is the formula for factorial?”, in Mathematics in School, volume 47, number 4, page 40:

(mathematics, combinatorics) The result of multiplying a given number of consecutive integers from 1 to the given number. In equations, it is symbolized by an exclamation mark (!). For example, 5! = 1 × 2 × 3 × 4 × 5 = 120.


permanent

From Middle English permanent, permanente, from Middle French permanent, from Latin permanēns, from permaneō (“I stay through”). First attested in the 15th century.

(linear algebra, combinatorics) Given an n×n matrix a_ij,, the sum over all permutations π, of ∏ᵢ₌₁ⁿa_iπ(i).


derangement

derangement (countable and uncountable, plural derangements)

Borrowed from French dérangement, equivalent to derange + -ment.

(mathematics, combinatorics) A permutation of a set such that no element is in its previous position.


binomial coefficient

binomial coefficient (plural binomial coefficients)

(combinatorics) a coefficient of any of the terms in the expansion of the binomial (x+y)ⁿ, defined by n choose k=(n!)/(k!(n-k)!), read as "n choose k"


greedoid

From greed (from its applicability to problems that can be solved by greedy algorithms) and matroid.

(combinatorics) A particular type of set system.


matroid

From matrix + -oid. They were so named by Hassler Whitney in 1935 in an article titled “On the abstract properties of linear dependence”.

A matroid can be defined in terms of bases. A matroid consists of a ground set as well as a set of bases which is a nonempty subset of the power set of the ground set.

(combinatorics) A structure that captures the essence of a notion of "independence" that generalizes linear independence in vector spaces and acyclicality in graphs.


coloop

(combinatorics) An element in a matroid that belongs to no circuit or (equivalently) belongs to every basis.


Dyck word

Named after the mathematician Walther von Dyck.

(mathematics, combinatorics) A string consisting of n X's and n Y's such that no initial segment of the string has more Y's than X's.


graphoid

(combinatorics) A matroid that has two collections of nonempty subsets called circuits and cocircuits such that the intersection of any circuit and cocircuit is not only the identity matroid, no circuit properly contains another circuit, no cocircuit properly contains another cocircuit, and for any painting of M that colors exactly one element green and the rest either red or blue there exists either a circuit containing the green element and no red elements or a cocircuit containing the green element and no blue elements.


double factorial

double factorial (plural double factorials)

The symbolical representation of the results of this paper is much facilitated by the introduction of a separate symbol for the product of alternate factors, n#92;cdotn-2#92;cdotn-4#92;cdots 1, if n be odd, or n#92;cdotn-2#92;cdots 2 if n be odd^([sic – meaning even]). I propose to write n#33;#33; for such products, and if a name be required for the product to call it the "alternate factorial" or the "double factorial." Full advantage of the new symbol is only gained by extending its meaning to the negative values of n. Its complete definition may then be included in the equationsn#33;#33;#61;n(n-2)#33;#33;,#92;quad 1#33;#33;#61;1,#92;quad 2#33;#33;#61;2.

1902, Arthur Schuster, “On some Definite Integrals and a New Method of reducing a Function of Spherical Co-ordinates to a Series of Spherical Harmonics”, in Proceedings of the Royal Society of London, volume 71, →DOI, →JSTOR, page 99:

The double factorial notation #92;begin#123;align#125;(2n)#33;#33;amp;#61;2#92;cdot 4#92;cdot 6#92;cdots(2n-2)(2n)#61;2#123;n#125;n#33;#92;#92;(2n#43;1)#33;#33;amp;#61;1#92;cdot 3#92;cdot 5#92;cdots(2n-1)(2n#43;1)#61;#92;frac#123;(2n#43;1)#33;#125;#123;2#123;n#125;n#33;#125;#92;end#123;align#125;may be considered as a generalization of n#33;#61;1#92;cdot 2#92;cdot 3#92;cdotsn.

1948 September, B. E. Meserve, “Double Factorials”, in The American Mathematical Monthly, volume 55, number 7, →DOI, →JSTOR, page 425:

We prefer now to write the expansion in a slightly different way in order to exhibit more clearly the symmetry properties of the expansion coefficients:f#95;k(m,m')#61;(#92;tfrac#123;1#125;#123;2#125;#92;pi)#92;tfrac#123;1#125;#123;2#125;2#123;-2#92;bar#123;m#125;-3#125;#92;sum#95;#123;#92;sigma#125;(2#92;bar#123;m#125;-2#92;sigma#43;1)#33;#33;#92;,T#95;k#123;#92;,2#92;sigma#125;(m,m')J#95;#123;2#92;sigma#125;,where, as above, #92;bar#123;m#125; is the average of m and m', #92;bar#123;m#125;#61;#92;tfrac#123;1#125;#123;2#125;(m#43;m'), and the double factorial notation is used, (2n#43;1)#33;#33;#61;1#92;cdot 3#92;cdot 5#92;cdots(2n#43;1).

1958–1959, Kenneth W. Ford, E. J. Konopinski, “Evaluation of Slater integrals with harmonic oscillator wave functions”, in Nuclear Physics, volume 9, number 2, →DOI, page 219:

The symbol #33;#33; in Eq. (9) denotes the double factorial given by n#33;#33;#61;n(n-2)(n-4)#92;dots#92;kappa#95;n, where #92;kappa#95;n is 1 for odd n and 2 for even n.

2008 April, Adriana Pálffy, Jörg Evers, Christoph H. Keitel, “Electric-dipole-forbidden nuclear transitions driven by super-intense laser fields”, in Physical Review C, volume 77, number 4, →DOI, page 044602-3:

Double factorials can also be defined recursively. Just as we can define the ordinary factorial by n#33;#61;n#92;cdot(n-1)#33; for n#92;geq 1 with 0#33;#61;1, we can define the double factorial byn#33;#33;#61;n#92;cdot(n-2)#33;#33;for n#92;geq 2 with initial values 0#33;#33;#61;1#33;#33;#61;1. With our convention that (-1)#33;#33;#61;1, the recursion is valid for all positive integers n.

2012 June, Henry Gould, Jocelyn Quaintance, “Double Fun with Double Factorials”, in Mathematics Magazine, volume 85, number 3, →DOI, pages 177–178:

(mathematics, combinatorics) For an odd integer, the result of multiplying all the odd integers from 1 to the given number; or for an even integer, the result of multiplying all the even integers from 2 to the given number; symbolised by a double exclamation mark (!!). For example, 9!! = 1 × 3 × 5 × 7 × 9 = 945.


Ramsey number

Named after Frank Plumpton Ramsey (1903–1930), British mathematician, philosopher and economist.

(combinatorics) Any one of a certain set of numbers which are guaranteed to exist by Ramsey's theorem; a positive integer which is a certain function of some given multiset of positive integers, where that "certain function" is that which yields the minimal number guaranteed to exist by Ramsey's theorem.


UST

UST (countable and uncountable, plural USTs)

(combinatorics) Initialism of uniform spanning tree.


subfactorial

(mathematics, combinatorics) The result of deranging a number. In equations, it is usually symbolised by an exclamation mark (!) before the number being deranged (as opposed to factorials, where the exclamation mark occurs after the number).


Helly family

Named after the mathematician Eduard Helly (1884-1943).

(combinatorics) A family of order k of sets, such that any minimal subfamily with an empty intersection has k or fewer sets in it.


excedance

A number i for which π(i) ≥ i is called a weak excedance of π, while a number i for which π(i) > i is called an excedance of π.

1997, Richard P. Stanley, Enumerative Combinatorics, 2nd edition, volume 1, page 23:

(mathematics, combinatorics) A position in a permutation occupied by a value greater than the position number.


difference set

(combinatorics) A subset of a group such that any element of the group can be expressed as the difference (or the quotient) of two elements in such subset.


Narayana number

Narayana number (plural Narayana numbers)

Named after Canadian mathematician T. V. Narayana (1930–1987).

(combinatorics) Any of the triangular array of natural numbers defined by N(n, k), n = 1, 2, 3 ..., 1 ≤ k ≤ n, occurring in various counting problems.


back substitution

back substitution (countable and uncountable, plural back substitutions)

(combinatorics) A method of solving recurrence relations.


bent function

So called because they are as different as possible from all linear functions (the simplest or "straight-line" functions) and from all affine functions (which preserve parallel lines).

(combinatorics) A Boolean function f: Z ₂ⁿ→ Z ₂ whose Walsh transform has constant absolute value.


Möbius function

Named after August Ferdinand Möbius (1790–1868), a German mathematician and theoretical astronomer who introduced it in 1832.

(combinatorics) A function from the natural numbers to the set {−1, 0, 1} which maps perfect squares to 0, prime numbers to −1, and is multiplicative.


Helly dimension

Helly dimension (plural Helly dimensions)

Helly's theorem implies that the Helly dimension of a Euclidean space equals its dimension as a real vector space.

(combinatorics) A value associated with a metric space, being one less than the Helly number of the family of metric balls in that space.


rook polynomial

rook polynomial (plural rook polynomials)

(combinatorics) A generating polynomial of the number of ways to place non-attacking rooks on a generalized form of chessboard.


Bell polynomial

Bell polynomial (plural Bell polynomials)

(combinatorics) Any member of a certain class of polynomials used in the study of set partitions, and related to Stirling numbers and Bell numbers.


Bell number

Named after Eric Temple Bell, who wrote about them in the 1930s.

(combinatorics) Any member of a sequence of numbers that count the possible partitions of a set, such that the nth number Bₙ is the number of different ways to partition a set that has exactly n elements.


de Bruijn torus

de Bruijn torus (plural de Bruijn toruses or de Bruijn tori)

de Bruijn toruses (plural)
de Bruijn tori (plural)

Named from the de Bruijn sequence, which can be considered a special case for n=1 (one dimension). It is a torus because the edges are considered to wrap around for the purpose of finding matrices.

(combinatorics) An array of symbols from an alphabet (often just 0 and 1) that contains every m-by-n matrix exactly once.


algebraic combinatorics

1982, Trevor Evans, Finite Representations of Two-variable Identities, E. Mendelsohn, Algebraic and Geometric Combinatorics, North-Holland, page 135,
It is part of the folklore of algebraic combinatorics that “most' two-variable groupoid identities have non-trivial models in finite fields, the groupoid operation being represented by a linear function ax+by.

This volume arose through the initiative of Kluwer Academic Publishers in an attempt to introduce some areas of research in algebraic combinatorics which originally appeared in Russian to a wider mathematical community.

1993, I. A. Faradžev, A. A. Ivanov, M. Klin, A. J. Woldar, editors, Investigations in Algebraic Theory of Combinatorial Objects, Kluwer Academic, page vii:

This book is intended primarily as a one-semester undergraduate text for a course in algebraic combinatorics.[…]Algebraic combinatorics is a huge subject, so some selection process was necessary to obtain the present text.

2013, Richard P. Stanley, Algebraic Combinatorics, 2nd edition, Springer, page xi:

(algebra, combinatorics) A branch of mathematics in which techniques from abstract algebra are applied to problems in combinatorics, and vice versa.

combinatoire algébrique

branch of mathematics at the intersection of combinatorics and algebra


jeu de taquin

From French jeu de taquin (“15 puzzle”, literally “teasing game”).

bor

equivalence relation
equivalence relation

(combinatorics) A construction that defines an equivalence relation on the set of skew standard Young tableaux.


jeu de taquin equivalent

en-adj

jeu de taquin equivalent (not comparable)

(combinatorics) Of tableaux: able to be transformed into each other by a sequence of jeu de taquin slides.


jeu de taquin slide

en-noun

jeu de taquin slide (plural jeu de taquin slides)

(combinatorics) A transformation that moves around the numbers in a tableau according to the jeu de taquin construction, used to determine whether tableaux are equivalent.


exact cover

A finite set #92;#123;S(#92;alpha#95;i,#92;beta#95;i)#92;vert 1#92;lei#92;lem#92;#125; is called an (eventual) exact cover if every (sufficiently large) positive integer occurs in exactly one S(#92;alpha#95;i,#92;beta#95;i). If #92;#123;S(#92;alpha#95;i,#92;beta#95;i)#92;#125;#95;#123;i#61;1#125;ᵐ is an eventual exact cover, then #92;textstyle#92;sum#95;#123;i#61;1#125;ᵐ#123;#92;alpha#95;i#123;-1

2000, R. Tijdeman, “Exact covers of balanced sequences and Fraenkel's conjecture”, in F. Halter-Koch, Robert F. Tichy, editors, Algebraic Number Theory and Diophantine Analysis: Proceedings of the International Conference, Walter de Gruyter, page 468:

2011, R. Lu, S. Liu, J. Zhang, Searching for Doubly Self-orthogonal Latin Squares, Jimmy Lee (editor), Principles and Practice of Constraint Programming: 17th International Conference CP 2011, Proceedings, Springer, LNCS 6876, page 542,
It is straightforward to use clique algorithms to construct a (partial) solution of a given combinatorial problem which is represented as a set system. If the solution of the combinatorial problem corresponds to the exact cover of the set system, a substantially more efficient algorithm can be utilized because of this property.

(topology, combinatorics) Given a collection S of subsets of a set X, a subcollection S^* of S such that each element of X is contained in exactly one subset in S^*.

exakte Überdeckung

Collection of subsets such that each element of the original set is contained in exactly one subset


semifactorial

With the symbol n#33;#33; we indicate the semifactorial of n, that is the product 1.3.5#92;dotsn if n is odd, and the product 2.4.6#92;dotsn if n is even.

1942 February, Giulio Racah, “Theory of Complex Spectra. I”, in Physical Review, volume 61, numbers 3–4, →DOI, page 188:

In the formulas above we use the Euler beta function, a Jacobi polynomial evaluated at three, and also the familiar double factorial (also known as the semifactorial) notation defined by n#33;#33;#61;#92;begin#123;cases#125;#92;displaystyle 1#92;times 3#92;times 5#92;times#92;dots#92;timesnamp;#92;text#123;if#125;n#92;text#123;isodd#125;,#92;#92;#91;5pt#93;#92;displaystyle 2#92;times 4#92;times 6#92;times#92;dots#92;timesnamp;#92;text#123;if#125;n#92;text#123;iseven#125;.#92;end#123;cases#125;By convention, 0#33;#33;#61;(-1)#33;#33;#61;1.

1994 January, Alan Edelman, Eric Kostlan, Michael Shub, “How many eigenvalues of a random matrix are real?”, in Journal of the American Mathematical Society, volume 7, number 1, →DOI, pages 250–251:

The double factorial or semifactorial function is defined byn#33;#33;#61;#92;begin#123;cases#125;#92;displaystyle 1amp;n#61;-1,0#92;#92;#91;5pt#93;#92;displaystylen#92;times(n-2)#92;times(n-4)#92;times#92;dots#92;times 5#92;times 3#92;times 1amp;n#61;1,3,5,#92;dots#92;#92;#91;5pt#93;#92;displaystylen#92;times(n-2)#92;times(n-4)#92;times#92;dots#92;times 6#92;times 4#92;times 2amp;n#61;2,4,6,#92;dots#92;end#123;cases#125;

2009, Keith B. Oldham, Jan C. Myland, Jerome Spanier, An Atlas of Functions, 2nd edition, Springer, →DOI, →ISBN, page 25:

double factorial
double factorial#English

(mathematics, combinatorics, rare) Synonym of double factorial


Sperner family

Sperner family (plural Sperner families)

(combinatorics) A family F of subsets of a finite set E in which none of the sets contains another; or, equivalently, an antichain in the inclusion lattice over the power set of E.


digraphoid

Blend of directed + graphoid, after digraph (“directed graph”). Coined by American mathematician George J. Minty in a 1966 article.

coinage

We now wish to consider what part of the theory of directed graphs can be built up in the wider context of graphoids/matroids. To this end, we shall introduce the concepts of digraphoid (short for “directed graphoid”) and orientable graphoid. […] A digraphoid is a structure consisting of: (1º) a graphoid, and (2º) a partitioning of each circuit and cocircuit of the graphoid, each being partitioned into two sets; this partitioning is to satisfy the axiom: […]

[1966, George J. Minty, “On the Axiomatic Foundations of the Theories of Directed Linear Graphs, Electrical Networks and Network-Programming”, in Journal of Mathematics and Mechanics, volume 15, number 3, →JSTOR, pages 506–507:

(combinatorics) A dual pair of regular matroids.