In mathematics, a vexillary permutation is a
permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or proc ...
''μ'' of the positive integers containing no
subpermutation isomorphic to the permutation (2143); in other words, there do not exist four numbers ''i'' < ''j'' < ''k'' < ''l'' with ''μ''(''j'') < ''μ''(''i'') < ''μ''(''l'') < ''μ''(''k''). They were introduced by . The word "vexillary" means flag-like, and comes from the fact that vexillary permutations are related to
flags
A flag is a piece of textile, fabric (most often rectangular or quadrilateral) with a distinctive design and colours. It is used as a symbol, a signalling device, or for decoration. The term ''flag'' is also used to refer to the graphic desi ...
of
modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a sy ...
.
showed that vexillary
involutions are enumerated by
Motzkin number
In mathematics, the th Motzkin number is the number of different ways of drawing non-intersecting chords between points on a circle (not necessarily touching every point by a chord). The Motzkin numbers are named after Theodore Motzkin and have d ...
s.
See also
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Riffle shuffle permutation In the mathematics of permutations and the study of shuffling playing cards, a riffle shuffle permutation is one of the permutations of a set of n items that can be obtained by a single riffle shuffle, in which a sorted deck of n cards is cut into t ...
, a subclass of the vexillary permutations
References
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*{{Citation , last1=Macdonald , first1=I.G. , author1-link=Ian G. Macdonald , title=Notes on Schubert polynomials , url=https://books.google.com/books?id=BvLuAAAAMAAJ , publisher=Laboratoire de combinatoire et d'informatique mathématique (LACIM), Université du Québec a Montréal , series=Publications du Laboratoire de combinatoire et d'informatique mathématique , isbn=978-2-89276-086-6 , year=1991b , volume=6
Permutation patterns