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In [[mathematics]], a '''differential poset''' is a [[partially ordered set]] (or ''poset'' for short) satisfying certain local properties. (The formal definition is given below.) This family of posets was introduced by {{harvtxt|Stanley|1988}} as a generalization of [[Young's lattice]] (the poset of [[integer partition]]s ordered by inclusion), many of whose [[combinatorics|combinatorial]] properties are shared by all differential posets. In addition to Young's lattice, the other most significant example of a differential poset is the [[
==Definitions==
A poset ''P'' is said to be a ''differential poset'', and in particular to be ''r''-''differential'' (where ''r'' is a positive [[integer]]), if it satisfies the following conditions:
* ''P'' is [[graded poset|graded]] and [[locally finite poset|locally finite]] with a unique minimal element;
* for every two distinct elements ''x'', ''y'' of ''P'', the number of elements [[covering relation|covering]] both ''x'' and ''y'' is the same as the number of elements covered by both ''x'' and
* for every element ''x'' of ''P'', the number of elements covering ''x'' is exactly ''r'' more than the number of elements covered by
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
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
==Examples==
[[File:Young-Fibonacci.svg|thumb|300px|The [[Young–Fibonacci graph]], the [[Hasse diagram]] of the Young–Fibonacci lattice.]]
The canonical examples of differential posets are [[Young's lattice]], the poset of [[integer partition]]s ordered by inclusion, and the
There is a canonical construction (called "reflection") of a differential poset given a finite poset that obeys all of the defining axioms below its top rank. (The
The Young-Fibonacci lattice has a natural ''r''-differential analogue for every positive integer
{{unsolved|mathematics|Are there any differential lattices that are not products of Young's lattice and the Young-Fibonacci lattices?}}▼
▲The Young-Fibonacci lattice has a natural ''r''-differential analogue for every positive integer ''r''. These posets are lattices, and can be constructed by a variation of the reflection construction. In addition, the product of an ''r''-differential and ''s''-differential poset is always an (''r'' + ''s'')-differential poset. This construction also preserves the lattice property. It is not known for any ''r'' > 1 whether there are any ''r''-differential lattices other than those than arise by taking products of the Young-Fibonacci lattices and Young's lattice.
▲{{unsolved|mathematics|Are there any differential lattices that are not products of Young's lattice and the
==Rank growth==
In addition to the question of whether there are other differential lattices, there are several long-standing open problems relating to the rank growth of differential posets. It was [[conjecture]]d in {{harvtxt|Stanley|1988}} that if ''P'' is a differential poset with {{math|''r''<sub>''n''</sub>}} vertices at rank ''n'', then
: <math>p(n) \le r_n \le F_n,</math>
where ''p''(''n'') is the [[partition function (number theory)|number of integer partitions]] of ''n'' and {{math|''F''<sub>''n''</sub>}} is the ''n''th [[Fibonacci number]]. In other words, the conjecture states that at every rank, every differential poset has a number of vertices lying between the numbers for Young's lattice and the Young-Fibonacci lattice. The upper bound was [[mathematical proof|proved]] in {{harvtxt|Byrnes|2012}}, while the lower bound remains open. {{harvtxt|Stanley|Zanello|2012}} proved an [[asymptotic analysis|asymptotic]] version of the lower bound, showing that
: <math> r_n \gg n^a \exp(2\sqrt{n}) </math>
for every differential poset and some constant ''a''. By comparison, the partition function has asymptotics
: <math> p(n) \sim \frac{1}{4n\sqrt{3}} \exp\left({\pi \sqrt {\frac{2n}{3}}}\right).</math>
All known bounds on rank sizes of differential posets are quickly growing functions. In the original paper of Stanley, it was shown (using [[eigenvalue]]s of the operator ''DU'') that the rank sizes are weakly increasing. However, it took 25 years before {{harvtxt|Miller|2013}} showed that the rank sizes of an {{nowrap|''r''-differential}} poset strictly increase (except trivially between ranks 0 and 1 when ''r'' = 1).
==Properties==
[[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''
* 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>{{hairsp}}''n''</sup>}}.{{sfn|Stanley|2011|p=385|loc=Theorem 3.21.8}}
* 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>{{hairsp}}''rx'' + ''x''<sup>2</sup>/2</sup>}}.{{sfn|Stanley|2011|p=386|loc=Theorem 3.21.10}}
==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|2007}} defined dual graded graphs corresponding to any [[Kac–Moody algebra]].
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.
==References==
{{Reflist}}
| year = 2007}} [http://math.umn.edu/~jblewis/docs/JBLHarvardSeniorThesis.pdf] ([[Harvard College]] undergraduate thesis)▼
== 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}}
▲
* {{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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