Timeline for Number of permutations in $S_{a+b}$ with $\operatorname{maj}(\pi)=a$ and $\operatorname{maj}(\pi^{-1})=b$
Current License: CC BY-SA 4.0
20 events
| when toggle format | what | by | license | comment | |
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| Feb 22, 2021 at 17:50 | vote | accept | Per Alexandersson | ||
| Feb 22, 2021 at 16:14 | answer | added | David E Speyer | timeline score: 3 | |
| Feb 22, 2021 at 7:28 | comment | added | Per Alexandersson | @T.Amdeberhan Right, that's correct. I was hoping that there's an easy argument to just compare coefficients, if one assumes $n$ is larger than or equal to $a+b$ | |
| Feb 21, 2021 at 22:17 | comment | added | T. Amdeberhan | @PerAlexandersson: You're probably aware that the RHS of the above formula for $S_{a+b}$ begins with $i,j=1$ while the latter formula begins with $i,j=0$. That said, the latter formula has extra product in the form $\prod\frac1{1-w^i}$ appearing twice which injects the generating function for integer partitions. Moreover, you know that the latter formula is a Cauchy product for Schur functions. | |
| Feb 21, 2021 at 21:57 | comment | added | T. Amdeberhan | @PerAlexandersson: You might also like to look into this paper: Don RAWLINGS, Generalized Worpitzky Identifies with Applications to Permutation Enumeration, Europ. J. Comb., Vol. 2, 1981, p. 67-78. (This is freely available online.) | |
| Feb 21, 2021 at 20:30 | comment | added | Per Alexandersson | @T.Amdeberhan Ah, yes, I could find it after some work - in remark 6.2, it is mentioned that the number of permutations in S_{r} with maj=a, imaj=b, with a,b<M, and r>2M, is independent of $r$. This is what's mentioned in the OEIS entry, that it stabilizes... | |
| Feb 21, 2021 at 20:01 | comment | added | Per Alexandersson | @T.Amdeberhan I also tried to find that reference, but it seems hard to access it. | |
| Feb 21, 2021 at 18:30 | comment | added | T. Amdeberhan | Have you tried this? M. S. Cheema and T. S. Motzkin, "Multipartitions and multipermutations," Proc. Symp. Pure Math. 19 (1971), 39-70, eq. (3.1.3).I don't have access here. | |
| Feb 21, 2021 at 18:17 | comment | added | LSpice |
P.S. I should say I like your TeX. I think not enough people make use of the semantic \dots. commands.
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| Feb 21, 2021 at 18:17 | comment | added | Per Alexandersson | @LSpice I appreciate the effort - our edits simply collided as I was adding the definition of major index. | |
| Feb 21, 2021 at 18:16 | comment | added | LSpice | @PerAlexandersson, certainly you are right, and I apologise for my error. I think I was applying the fix as you were converting it back to the static form. Anyway I try to edit with a light touch, and the preview seemed to confirm that everything was working, so I apologise that in the end it wound up being ugly. | |
| Feb 21, 2021 at 18:06 | history | edited | LSpice | CC BY-SA 4.0 |
Deleted leading space due to carriage return
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| Feb 21, 2021 at 18:05 | comment | added | LSpice |
@PerAlexandersson, I see you prefer not to use \label and \eqref. I apologise for editing in something you didn't want. The main advantage is that it makes the references clickable links to the equations. Obviously in a short post rather than a paper it's not a big deal, but it's a nice affordance! (Or maybe you saw revision 2, where the ugly rendered result didn't match the preview. I fixed that in revision 3, and filed a bug report.)
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| Feb 21, 2021 at 18:00 | history | edited | Per Alexandersson | CC BY-SA 4.0 |
wth?
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| Feb 21, 2021 at 17:59 | history | edited | LSpice | CC BY-SA 4.0 |
\eqref worked in preview but not when posted, sorry!
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| Feb 21, 2021 at 17:59 | comment | added | LSpice | Major index. | |
| Feb 21, 2021 at 17:50 | comment | added | Wojowu | What is $\mathrm{maj}$? | |
| Feb 21, 2021 at 17:45 | history | edited | LSpice | CC BY-SA 4.0 |
Mild TeXing
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| Feb 21, 2021 at 17:34 | comment | added | Turbo | $maj(\pi)$ is $\mbox{definition}$? | |
| Feb 21, 2021 at 17:32 | history | asked | Per Alexandersson | CC BY-SA 4.0 |