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...[Ideal (order theory)|downset]] in the permutation pattern order.{{r|k}} A permutation class may also be known as a '''pattern class''', '''closed class''', or si ...classes have bases with more than one pattern or even with infinitely many patterns. ...

In the mathematics of [[permutation]]s, a '''layered permutation''' is a permutation that reverses contiguous blocks of elements. Equivalently, it is the [[Skew ...Stanley–Wilf conjecture]] for classes of permutations forbidding a layered permutation, before the conjecture was proven more generally.{{r|bona}} ...

...theory of [[permutation pattern]]s, a '''skew-merged permutation''' is a [[permutation]] that can be partitioned into an increasing sequence and a decreasing sequ ...tion can also be equivalently defined as a permutation that avoids the two patterns 3412 and 2143. ...

...permutation ''σ'' of length ''n'', the skew sum of ''π'' and ''σ'' is the permutation of length ''m'' + ''n'' defined by and the direct sum of ''π'' and ''σ'' is the permutation of length ''m'' + ''n'' defined by ...

...d set (1 rising sequence), mathematically a riffle shuffle is defined as a permutation on this set containing 1 or 2 rising sequences.<ref>{{citation ...o <math>n</math> (which would be <math>2^n</math>), because the [[identity permutation]] can be represented in multiple ways as a <math>(p,q)</math>-shuffle for d ...

...reserved by this type of shuffle, and a '''Gilbreath permutation''' is a [[permutation]] that can be formed by a Gilbreath shuffle.<ref name="dg">{{citation|title ...ern]]s, the Gilbreath permutations are the permutations that avoid the two patterns 132 and 312.<ref name="v">{{citation ...

...]] <math>\sigma \in S_n</math> which satisfies the following generalized [[Permutation pattern|pattern avoidance]] property: ...incular pattern]]s, a Baxter permutation is one that avoids the two dashed patterns <math>2-41-3</math> and <math>3-14-2</math>. ...

In [[combinatorics|combinatorial mathematics]], a '''partial permutation''', or '''sequence without repetition''', on a [[finite set]] ''S'' ...alently, it is a [[partial function]] on ''S'' that can be extended to a [[permutation]].<ref>{{citation ...

...rtable permutations are exactly the permutations that do not contain the [[permutation pattern]] 231; they are counted by the [[Catalan number]]s, and may be plac ...orrectly sorts as being exactly the permutations that do not contain the [[permutation pattern]] 231: three elements ''x'', ''y'', and ''z'', appearing in the inp ...

...tions of length <math>n=4</math> generated by Heap's algorithm, where each permutation is color-coded (1=blue, 2=green, 3=yellow, 4=red).]] ...y computer.<ref>{{Cite journal | last1 = Sedgewick | first1 = R. | title = Permutation Generation Methods | doi = 10.1145/356689.356692 | journal = ACM Computing ...

...he number of (132)-avoiding permutations that have a given number of (123) patterns"<ref name=":1" /> with the result being "in the form of a continued fractio ...

Any permutation of the corners is possible, including odd permutations. Seven of the corner ...nguishable, since corresponding edges are mirror images of each other. Any permutation of the edges in each set is possible, including odd permutations, giving 24 ...

...nexpected instances of [[Wilf equivalence]], where two seemingly-unrelated permutation classes have the same numbers of permutations of each length. [[stack-sortable permutation|231]] ...

...ation to the sequence 123...; for instance the sequence 213 represents the permutation on three elements that swaps elements 1 and 2. If π and σ are two permutat ...bsp;''x''&nbsp;<&nbsp;''z'', the same as the ordering of the values in the permutation 213. ...

...dual nature of community reassembly in California streams: a multivariate permutation-based approach|journal=Global Change Biology|volume=17|year=2011|issue=12|p ...

...a index]]) of a [[complete graph]] on {{mvar|n}} vertices, the number of [[permutation]]s on {{mvar|n}} elements that are [[involution (mathematics)|involution]]s ...e additional pattern in which no calls are being made, for a total of four patterns.<ref>{{citation ...

...ess-date=19 December 2023}}</ref> using 24 unique [[square]]s with 3-color patterns; each of the four edges is assigned a single color. The complete set of 24 ...hapes with <math>n</math> colors, MacMahon determined the number of unique patterns are: ...

...[mathematics]], the '''transposable integers''' are integers that [[cyclic permutation|permute]] or [[cyclic shift|shift]] cyclically when they are multiplied by ...te, the intention of the above expression is to show that the six [[cyclic permutation]]s of 006993 can be obtained from this repeating decimal if we select six c ...

Any permutation of the corners is possible, including odd permutations. Seven of the corner ...e the orientation of the twelfth edge depends on the preceding eleven. Any permutation of the intermediate and outer edges is possible, including odd permutations ...

...ith the first occurrences of each number in sorted order) that avoid the [[permutation pattern]]s 12312 and 121323.<ref>{{citation | title = Matchings avoiding partial patterns ...