Timeline for a Vandermonde-type of determinants summed over permutations
Current License: CC BY-SA 4.0
9 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 28, 2021 at 15:29 | comment | added | Max Alekseyev | @FanGe: Ok, added this to the answer. | |
| Jul 28, 2021 at 15:29 | history | edited | Max Alekseyev | CC BY-SA 4.0 |
added explanation
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| Jul 27, 2021 at 22:38 | comment | added | Fan Ge | @Alekseyev. I wasn't asking why but was suggesting a clarification in your answer. But anyway, very nice observation! | |
| Jul 27, 2021 at 20:41 | comment | added | Max Alekseyev | @FanGe: Only such terms do not vanish under alternating summation over $\sigma$. Furthermore, the summation of each such term gives $V$ multiplied by the sign of the term multi-degree. The question on whether $c(n)\ne 0$ seems to be hard. | |
| Jul 27, 2021 at 19:53 | comment | added | Fan Ge | @Petrov. Neat expression! | |
| Jul 27, 2021 at 19:48 | comment | added | Fan Ge | @Alekseyev. Yes, I agree. Maybe you can add a one-sentence explanation (e.g., this follows from looking at the coefficient of $a^{(0,1,...,n-1)}$ in each summand in the definition of $D$.) But still, is $c(n)$ non-zero for all $n$? | |
| Jul 27, 2021 at 5:07 | comment | added | Fedor Petrov | alternatively, $c(n)=[x_1^{n-1}x_2^{n-1}\ldots x_n^{n-1}] V(x_n,x_{n-1},\ldots,x_1)V(x_1,x_2-x_1,\ldots,x_n-x_{n-1})$, where $V(x_1,\ldots,x_n)=\prod_{i<j} (x_j-x_i)$. | |
| Jul 26, 2021 at 21:42 | history | edited | Max Alekseyev | CC BY-SA 4.0 |
degree's corrected
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| Jul 26, 2021 at 21:27 | history | answered | Max Alekseyev | CC BY-SA 4.0 |