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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==
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