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These basic properties may be restated in various ways. For example, Stanley shows that the number of elements covering two distinct elements ''x'' and ''y'' of a differential poset is always either 0 or 1, so the second defining property could be altered accordingly.
The defining properties may also be restated in the following [[linear algebra]]ic setting: taking the elements of the poset ''P'' to be formal [[basis (linear algebra)|basis]] vectors of an (infinite-dimensional) [[vector space]], let ''D'' and ''U'' be the [[linear operator|operators]] defined so that ''D'' ''x'' is equal to the sum of the elements covered by ''x'', and ''U'' ''x'' is equal to the sum of the elements covering ''x''. (The operators ''D'' and ''U'' are called the ''down'' and ''up operator'', for obvious reasons.) Then the second and third conditions may be replaced by the statement that ''DU'' − ''UD'' = ''r''
This latter reformulation makes a differential poset into a combinatorial realization of a [[Weyl algebra]], and in particular explains the name ''differential'': the operators "[[derivative|''d''/''dx'']]" and "multiplication by ''x''" on the vector space of [[polynomial]]s obey the same commutation relation as ''U'' and ''D''/''r''.
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[[File:Young's lattice.svg|thumb|300px|A Hasse diagram of Young's lattice]]
Every differential poset ''P'' shares a large number of combinatorial properties. A few of these include:
* The number of paths of length 2''n'' in the Hasse diagram of ''P'' beginning and ending at the minimal element is {{math|(2''n'' − 1)!!}}, where {{math|''m''!!}} is the [[double factorial]] function. In an {{nowrap|''r''-differential}} poset, the number of such paths is {{math|(2''n'' − 1)!! ''r''<sup>
* The number of paths of length 2''n'' in the Hasse diagram of ''P'' beginning with the minimal element such that the first ''n'' steps are covering relations from a smaller to a larger element of ''P'' while the last ''n'' steps are covering relations from a larger to a smaller element of ''P'' is {{math|''n''!}}. In an {{nowrap|''r''-differential}} poset, the number is {{math|''n''! ''r''<sup>
* The number of upward paths of length ''n'' in the Hasse diagram of ''P'' beginning with the minimal element is equal to the number of [[involution (mathematics)|involutions]] in the [[symmetric group]] on ''n'' letters. In an {{nowrap|''r''-differential}} poset, the sequence of these numbers has [[exponential generating function]] {{math|''e''<sup>
==Generalizations==
In a differential poset, the same set of edges is used to compute the up and down operators ''U'' and ''D''. If one permits different sets of up edges and down edges (sharing the same vertex sets, and satisfying the same relation), the resulting concept is the ''dual graded graph'', initially defined by {{harvtxt|Fomin|1994}}. One recovers differential posets as the case that the two sets of edges coincide.
Much of the interest in differential posets is inspired by their connections to [[representation theory]]. The elements of Young's lattice are integer partitions, which encode the [[representations of the symmetric group]]s, and are connected to the [[ring of symmetric functions]]; {{harvtxt|Okada|1994}} defined [[algebra over a field|algebras]] whose representation is encoded instead by the Young–Fibonacci lattice, and allow for analogous constructions such as a Fibonacci version of symmetric functions. It is not known whether similar algebras exist for every differential poset.{{citation needed|date=May 2017}} In another direction, {{harvtxt|Lam|Shimozono|
Other variations are possible; {{harvtxt|Stanley|1990}} defined versions in which the number ''r'' in the definition varies from rank to rank, while {{harvtxt|Lam|2008}} defined a signed analogue of differential posets in which cover relations may be assigned a "weight" of −1.
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== Sources ==
* {{citation |last=Stanley |first=Richard |year=2011 |title=Enumerative Combinatorics |edition=2 |volume=1 |url=http://www-math.mit.edu/~rstan/ec/ec1.pdf |access-date=2015-09-21 |archive-date=2011-05-31 |archive-url=https://web.archive.org/web/20110531113533/http://www-math.mit.edu/~rstan/ec/ec1.pdf |url-status=dead }}
* {{citation |last=Byrnes |first=Patrick |title=Structural Aspects of Differential Posets |year=2012 |isbn=9781267855169}}
* {{citation |last=Fomin |first=Sergey |author-link=Sergey Fomin |title=Duality of graded graphs |journal=Journal of Algebraic Combinatorics |volume=3 |year=1994 |issue=4 |pages=357–404 |doi=10.1023/A:1022412010826 |doi-access=free}}
* {{citation |last1=Lam |first1=Thomas F. |title=Signed differential posets and sign-imbalance |journal=Journal of Combinatorial Theory, Series A |volume=115 |issue=3 |year=2008 |pages=466–484 |doi=10.1016/j.jcta.2007.07.003 |arxiv=math/0611296 |s2cid=10802016}}
* {{citation |last1=Lam |first1=Thomas F. |last2=Shimozono |first2=Mark |title=Dual graded graphs for Kac-Moody algebras |journal=Algebra & Number Theory |volume=1 |year=2007 |issue=4 |pages=451–488 |doi=10.2140/ant.2007.1.451 |arxiv=math/0702090 |s2cid=18253442}}
▲ | url = https://blogs.gwu.edu/jblewis/files/2018/04/JBLHarvardSeniorThesis-20drube.pdf}} ([[Harvard College]] undergraduate thesis)
* {{citation |last=Miller |first=Alexander |year=2013 |title=Differential posets have strict rank growth: a conjecture of Stanley |journal=Order |volume=30 |issue=2 |pages=657–662 |doi=10.1007/s11083-012-9268-y |arxiv=1202.3006 |s2cid=38737147}}
* {{citation |last=Okada |first=Soichi |title=Algebras associated to the Young-Fibonacci lattice |journal=Transactions of the American Mathematical Society |volume=346 |issue=2 |pages=549–568 |year=1994 |publisher=American Mathematical Society |doi=10.2307/2154860 |jstor=2154860 |doi-access=free}}
*{{citation |last=Stanley |first=Richard P. |author-link=Richard P. Stanley |title=Differential posets |journal=Journal of the American Mathematical Society |volume=1 |issue=4 |pages=919–961 |year=1988 |doi=10.2307/1990995 |jstor=1990995 |publisher=American Mathematical Society |doi-access=free}}
* {{citation |last=Stanley |first=Richard P. |author-link=Richard P. Stanley |chapter=Variations on differential posets |title=Invariant theory and tableaux (Minneapolis, MN), 1988 |series=IMA Vol. Math. Appl. |volume=19 |pages=145–165 |publisher=Springer |year=1990}}
* {{citation |last1=Stanley |first1=Richard P. |author1-link=Richard P. Stanley |last2=Zanello |first2=Fabrizio |title=On the Rank Function of a Differential Poset |journal=Electronic Journal of Combinatorics |volume=19 |issue=2 |year=2012 |pages=P13 |doi=10.37236/2258 |s2cid=7405057 |url=http://www.combinatorics.org/ojs/index.php/eljc/article/view/v19i2p13 |doi-access=free|arxiv=1111.4371 }}
[[Category:Representation theory]]
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