|
<math display=block>{k+\ell \choose \ell}_q = {k+\ell \choose k}_q = \frac{\prod^{k+\ell}_{j=1}(1-q^j)}{\prod^{k}_{j=1}(1-q^j)\prod^{\ell}_{j=1}(1-q^j)}.</math>
The Gaussian binomial coefficient is related to the [[generating function]] of {{math|''p''(''N'', ''M''; ''n'')}} by the equality
=== <math display= "block ">\sum^{MN}_{n=0}p(N,M;n)q^n = {M+N \choose M}_q.</math> ===▼
▲=== <math display="block">\sum^{MN}_{n=0}p(N,M;n)q^n = {M+N \choose M}_q.</math> ===
=== Super distinct partitions ===
A super distinct partition if every difference is at least 2. For example, there are seven super distinct partitions of 11: <nowiki>{{11}, {10,1},{9,2},{8,3},{7,4},{7,3,1},{6,4,1}}</nowiki>. The partitions that with distinct parts that are not super distinct are <nowiki>{{8,2,1},{6,5},{6,3,2},{5,4,2},{5,3,2,1}}</nowiki>. The number of partitions of n into super distinct partitions is equivalent to the number of partitions of n into distinct parts such that every even part has a value greater than 2 times the number of odd parts present. This can be written in partition notation as p(n | super-distinct parts) = p(n | distinct parts, every even part > 2 (number of odd parts)). For n=11, we would have <nowiki>{{11},{10,1},{8,3},{7,4},{7,3,1},{6,4,1},{6,5}}</nowiki>. David Bressoud presented a proof of a bijection between the super-distinct partitions and the partitions such that every even part is greater than 2 times the number of odd parts in 1980.<ref>{{Cite web |title=How to make a function that returns all super distinct partitions? |url=https://mathematica.stackexchange.com/questions/288395/how-to-make-a-function-that-returns-all-super-distinct-partitions |access-date=2023-08-17 |website=Mathematica Stack Exchange |language=en}}</ref> Another example is 54's super distinct partition {17,15,12,8,2} corresponds to {16,12,11,9,6}.<ref>{{Cite web |title=Slideplayer presentation |url=http://player.slideplayer.com/download/10/2772378/g2qgMmjeKt8WgGBgww8Hdg/1691086330/2772378.ppt}}</ref><ref>{{Cite book |last=Andrews |first=George |title=Integer Partitions |last2=Eriksson |first2=Kimmo |publisher=Cambridge University Press |year=2004 |isbn=978-0-521-84118-4 |pages=23}}</ref><ref>{{Cite web |title=A003114 - OEIS |url=https://oeis.org/A003114 |access-date=2023-08-17 |website=oeis.org}}</ref>
==Rank and Durfee square==
|
