Talk:Q-Pochhammer symbol

q-binomial theorem?

As much as I can find, the wikilink to q-binomial theorem (which is the Gaussian binomial coefficient article), does not contain a discussion or the statement of the q-binomial theorem. 69.142.88.160 (talk) 15:13, 27 September 2020 (UTC)

Thanks for pointing this out. The target actually did contain the relevant content, it was just not labeled as such (and so almost useless for someone who didn't already know what they were looking for). I have added a subsection header in the relevant place and made the redirect point there -- hopefully this helps. --JBL (talk) 17:11, 27 September 2020 (UTC)

q-analog?

Can anybody explain the q-analog claim in the opening sentence? As far as I've seen, and according to the Q-analog article, a q-analog should give a classical notion for q=1. However here I get $(a;1)_{n}=\prod _{k=0}^{n-1}(1-a)=(1-a)^{n}$ which is not like a Pochhammer symbol at all (in any of its meanings). Marc van Leeuwen (talk) 13:19, 4 December 2009 (UTC)

Recent edits

‎The actual King of France recently made a large edit to the article, which I reworked. I would be happy to discuss those edits (and why the addition required reworking) here, but simply reverting is not appropriate. --JBL (talk) 17:05, 18 May 2018 (UTC)

I think it is not appropriate to just delete a contribution without providing any reasons, as you did. The factorials do not correspond to flags. I was inserting now a reference. You can read, for example https://arxiv.org/pdf/1006.2193.pdf or the famous article of Goldman & Rota in the binomial case. The analogy between sequences and flags was not explained at all in this article. I wrote an explanation that now has been replaced by something obscure: could you explain what is the "equivalence" between permutations and flags of sets? moreover, I do not see any need to introduce the notation $[n]!_{q}$ if the notation $[n]_{q}!$ is already used in the article Q-analog and in standard references as Kac & Cheung, "Quantum calculus". I analysed all the changes you made and for me they just seem unnecesary or simply wrong. The actual King of France (talk) 17:17, 18 May 2018 (UTC)

The characterization "just deleting a contribution" does not describe my edit even slightly: I took the material you added and reordered and reintegrated it into the section. (As you can see in the article history, your edit added 1297 bytes, mine removed only 108.) By contrast, your edit reverted my contribution without comment! So perhaps you should be more careful about your complaints.

On the specific substantive issues: of course the q-factorial counts flags. Since you like the q-multinomial, it is enough to note that

$[n]!_{q}=\left[{n \atop 1,\ldots ,1}\right]_{q}$

and so it counts complete flags, just as I wrote. Similarly, the equivalence between permutations and flags of sets is completely straightforward: the permutation tells you the orders in which the elements get added, it is exactly a special case of the more complicated thing you added. About the two notations, both are widespread, this one is obviously better, but I'm not going to go to war about it. --JBL (talk) 17:58, 18 May 2018 (UTC)

I'm sorry, I agree I was not fair with your contribution.

So, let's focus on the content: I get the point that in particular the counting of certain flags can give $[n]!_{q}$ / $[n]_{q}!$. But either you present this as a particular case of a more elaborate reasoning with the multinomial coefficients (as you're doing above) or you include a specific argument to justify the claim (that will be just a particular case of the more general one). Therefore, I still think it can be more clarifying to state an explicit correspondence between sequences and flags of sets, to provide the link to a reference about the counting of flags (e.g. Prasad) and then deduce the correspondence you're talking about as a particular case. The actual King of France (talk) 14:31, 19 May 2018 (UTC)

@The actual King of France: thanks, it is always pleasant to talk collaboratively with someone. Let me lay out the thinking behind my edits, and then invite you to be the one to make the next changes: I feel that, typically, the q-factorial and q-binomial coefficient are simpler and come up much more than more general multinomial. And of course q-binomial plays a particularly important role as far as Pochhammer symbols are concerned because of the q-binomial theorem. So it is true that one could state results for the general multinomial and then specialize, but (it seems to me that) for expository purposes it is better to dwell on the special cases first, then introduce the general case. Maybe it would help to explicitly mention in the article exactly how the factorial/complete flag situation is a special case?

I've added citations for both the claims about (complete) flags of sets and of vector spaces (actually, the same citation twice). A citation for the binomial and multinomial stuff would also be good. (I mean, I'm sure I can find it in Stanley, too.) Explaining the connection between permutations and flags-of-sets is a good suggestion; another possibility would be to not mention permutations and just assert that n! counts flags of sets (so no equivalence needed). I invite you to have a go at it (or I can, if you prefer). --JBL (talk) 20:46, 19 May 2018 (UTC)

q-factorial series coefficients

In the series expansion

$[n]_{q}!=\sum _{k=0}^{n(n-1)/2}a(k,n)q^{k}$

the coefficients $a(k,n)$ show up as the number of permutations of k items out of n. Is there an explicit name for these? The section Permutation#Inversions gives them the cryptic name Mahonian number but that article is silent on the matter. Anything clever to be said about them? Is there an explicit formula, besides the obvious brute-force just crank-on-it-and-multiply-it-out? 67.198.37.16 (talk) 23:29, 5 January 2024 (UTC)

the number of permutations of k items out of n no: the number of permutations of n objects that have k inversions. There is no formula for these numbers that is more useful than their generating function (which also encodes a simple recurrence relation). There's a lot of research into other permutation statistics that have the same distribution, analogues in other settings related to permutations, ..., and the asymptotics of the distribution are known, but I don't think there's a lot to say about the Mahonian distribution as a collection of numbers per se. --JBL (talk) 17:02, 7 January 2024 (UTC)