Editing Double factorial (section)

==Applications in enumerative combinatorics==
[[File:Unordered binary trees with 4 leaves.svg|thumb|300px|The fifteen different [[rooted binary tree]]s (with unordered children) on a set of four labeled leaves, illustrating {{math|15 {{=}} (2 × 4 − 3)‼}} (see article text).]]
Double factorials are motivated by the fact that they occur frequently in [[enumerative combinatorics]] and other settings. For instance, {{math|''n''‼}} for odd values of {{mvar|n}} counts
*[[Perfect matching]]s of the [[complete graph]] {{math|''K''<sub>''n'' + 1</sub>}} for odd {{mvar|n}}. In such a graph, any single vertex ''v'' has {{mvar|n}} possible choices of vertex that it can be matched to, and once this choice is made the remaining problem is one of selecting a perfect matching in a complete graph with two fewer vertices. For instance, a complete graph with four vertices ''a'', ''b'', ''c'', and ''d'' has three perfect matchings: ''ab'' and ''cd'', ''ac'' and ''bd'', and ''ad'' and ''bc''.<ref name="callan"/> Perfect matchings may be described in several other equivalent ways, including [[Involution (mathematics)|involutions]] without fixed points on a set of {{math|''n'' + 1}} items ([[permutations]] in which each cycle is a pair)<ref name="callan"/> or [[Chord diagram (mathematics)|chord diagrams]] (sets of chords of a set of {{math|''n'' + 1}} points evenly spaced on a circle such that each point is the endpoint of exactly one chord, also called [[Richard Brauer|Brauer]] diagrams).<ref name="dm93"/><ref>{{cite book|title=Patterns in Permutations and Words|series=EATCS Monographs in Theoretical Computer Science |first=Sergey|last=Kitaev| author-link = Sergey Kitaev|publisher=Springer|year=2011|isbn=9783642173332|page=96|url=https://books.google.com/books?id=JgQHtgR5N60C&pg=PA96}}</ref><ref>{{cite journal
 | last1 = Dale | first1 = M. R. T.
 | last2 = Narayana | first2 = T. V.
 | doi = 10.1016/0378-3758(86)90161-8
 | issue = 2
 | journal = Journal of Statistical Planning and Inference
 | mr = 852528
 | pages = 245–249
 | title = A partition of Catalan permuted sequences with applications
 | volume = 14
 | year = 1986}}</ref> The numbers of matchings in complete graphs, without constraining the matchings to be perfect, are instead given by the [[Telephone number (mathematics)|telephone number]]s, which may be expressed as a summation involving double factorials.<ref>{{cite journal
 | last1 = Tichy | first1 = Robert F.
 | last2 = Wagner | first2 = Stephan
 | doi = 10.1089/cmb.2005.12.1004
 | issue = 7
 | journal = [[Journal of Computational Biology]]
 | pages = 1004–1013
 | title = Extremal problems for topological indices in combinatorial chemistry
 | url = http://www.math.tugraz.at/fosp/pdfs/tugraz_main_0052.pdf
 | volume = 12
 | year = 2005| pmid = 16201918}}</ref>
*[[Stirling permutation]]s, permutations of the [[multiset]] of numbers {{math|1, 1, 2, 2, ..., {{mvar|k}}, {{mvar|k}}}} in which each pair of equal numbers is separated only by larger numbers, where {{math|''k'' {{=}} {{sfrac|''n'' + 1|2}}}}. The two copies of {{mvar|k}} must be adjacent; removing them from the permutation leaves a permutation in which the maximum element is {{math|''k'' − 1}}, with {{mvar|n}} positions into which the adjacent pair of {{mvar|k}} values may be placed. From this recursive construction, a proof that the Stirling permutations are counted by the double permutations follows by induction.<ref name="callan"/> Alternatively, instead of the restriction that values between a pair may be larger than it, one may also consider the permutations of this multiset in which the first copies of each pair appear in sorted order; such a permutation defines a matching on the {{math|2''k''}} positions of the permutation, so again the number of permutations may be counted by the double permutations.<ref name="dm93">{{cite journal
 | last1 = Dale | first1 = M. R. T.
 | last2 = Moon | first2 = J. W.
 | doi = 10.1016/0378-3758(93)90035-5
 | issue = 1
 | journal = Journal of Statistical Planning and Inference
 | mr = 1209991
 | pages = 75–87
 | title = The permuted analogues of three Catalan sets
 | volume = 34
 | year = 1993}}</ref>
*[[Heap (data structure)|Heap-ordered tree]]s, trees with {{math|''k'' + 1}} nodes labeled {{math|0, 1, 2, ... {{mvar|k}}}}, such that the root of the tree has label 0, each other node has a larger label than its parent, and such that the children of each node have a fixed ordering. An [[Euler tour technique|Euler tour]] of the tree (with doubled edges) gives a Stirling permutation, and every Stirling permutation represents a tree in this way.<ref name="callan"/><ref>{{cite conference
 | last = Janson | first = Svante | author-link = Svante Janson
 | arxiv = 0803.1129
 | contribution = Plane recursive trees, Stirling permutations and an urn model
 | mr = 2508813
 | pages = 541–547
 | publisher = Assoc. Discrete Math. Theor. Comput. Sci., Nancy
 | series = Discrete Math. Theor. Comput. Sci. Proc., AI
 | title = Fifth Colloquium on Mathematics and Computer Science
 | year = 2008| bibcode = 2008arXiv0803.1129J}}</ref>
*[[Unrooted binary tree]]s with {{math|{{sfrac|''n'' + 5|2}}}} labeled leaves. Each such tree may be formed from a tree with one fewer leaf, by subdividing one of the {{mvar|n}} tree edges and making the new vertex be the parent of a new leaf.
*[[Rooted binary tree]]s with {{math|{{sfrac|''n'' + 3|2}}}} labeled leaves. This case is similar to the unrooted case, but the number of edges that can be subdivided is even, and in addition to subdividing an edge it is possible to add a node to a tree with one fewer leaf by adding a new root whose two children are the smaller tree and the new leaf.<ref name="callan"/><ref name="dm93"/>

{{harvtxt|Callan|2009}} and {{harvtxt|Dale|Moon|1993}} list several additional objects with the same [[combinatorial class|counting sequence]], including "trapezoidal words" ([[numeral system|numerals]] in a [[mixed radix]] system with increasing odd radixes), height-labeled [[Dyck path]]s, height-labeled ordered trees, "overhang paths", and certain vectors describing the lowest-numbered leaf descendant of each node in a rooted binary tree. For [[bijective proof]]s that some of these objects are equinumerous, see {{harvtxt|Rubey|2008}} and {{harvtxt|Marsh|Martin|2011}}.<ref>{{cite conference
 | last = Rubey | first = Martin
 | contribution = Nestings of matchings and permutations and north steps in PDSAWs
 | mr = 2721495
 | pages = 691–704
 | publisher = Assoc. Discrete Math. Theor. Comput. Sci., Nancy
 | series = Discrete Math. Theor. Comput. Sci. Proc., AJ
 | title = 20th Annual International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2008)
 | year = 2008}}</ref><ref>{{cite journal
 | last1 = Marsh | first1 = Robert J.
 | last2 = Martin | first2 = Paul
 | arxiv = 0906.0912
 | doi = 10.1007/s10801-010-0252-6
 | issue = 3
 | journal = [[Journal of Algebraic Combinatorics]]
 | mr = 2772541
 | pages = 427–453
 | title = Tiling bijections between paths and Brauer diagrams
 | volume = 33
 | year = 2011| s2cid = 7264692
 }}</ref>

The even double factorials give the numbers of elements of the [[hyperoctahedral group]]s (signed permutations or symmetries of a [[hypercube]])