Timeline for Parameter independence of Stanley's "content formula". Why?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Mar 14, 2021 at 19:26 | comment | added | lambda | @InesInstitoris I'm not entirely sure why you're asking me specifically, but I don't see an obvious reason you'd get such a characterization. | |
| Mar 14, 2021 at 17:56 | comment | added | Jeanne Scott | @lambda Sorry to respond to your question with a question, but does this invariance (i.e. the independence with respect to $q$) lead to a characterization of the content statistic ? | |
| Mar 14, 2021 at 17:42 | comment | added | lambda | @T.Amdeberhan I guess it's up to you whether you think it answers the question or not, but I left it as a comment and not an answer because I assumed not. It's a combinatorial/conceptual explanation of why the first formula depends only on $tv$, but doesn't readily re-express itself in terms of $q$ and $a$. | |
| Mar 14, 2021 at 16:01 | comment | added | T. Amdeberhan | @lambda: It is not clear if you're answering the question or just remarking. | |
| Mar 14, 2021 at 2:26 | comment | added | lambda | The RHS is the exponential generating function for permutations by number of cycles. By equation (4) in the paper, $t$ counts cycles of $\sigma$ while $v$ counts cycles of $\sigma^{-1}$, but those are the same so you just get powers of $tv$. | |
| Mar 13, 2021 at 23:29 | comment | added | Sam Hopkins | Sorry, you're right, transposition shows that the expression has to be invariant under $q \mapsto -q$, but not that it is independent of $q$ altogether. | |
| Mar 13, 2021 at 23:24 | comment | added | T. Amdeberhan | If $f$ is a polynomial, then $f(x)+f(-x)$ has no odd-power terms (including $x$ term). That doesn't mean $f$ has no $x$ term. Does it? | |
| Mar 13, 2021 at 22:29 | comment | added | Sam Hopkins | Doesn’t transposition cancel out all the terms involving only the first power of the content? | |
| Mar 13, 2021 at 22:27 | history | edited | T. Amdeberhan | CC BY-SA 4.0 |
edited title
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| Mar 13, 2021 at 22:00 | history | edited | LSpice | CC BY-SA 4.0 |
Name of Stanley's paper
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| Mar 13, 2021 at 21:54 | history | asked | T. Amdeberhan | CC BY-SA 4.0 |