Timeline for Combinatorial proof of a certain binomial identity
Current License: CC BY-SA 4.0
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| Mar 10, 2023 at 21:57 | history | edited | Mikhail Tikhomirov | CC BY-SA 4.0 |
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| Mar 10, 2023 at 21:46 | comment | added | Mikhail Tikhomirov | I'd like to point a different cute interpretation of $(\heartsuit)$, divided by $4^{n+ p + q} {n + p + q \choose n}$. Define $X_{n, t}$ as the following random variable: let $W$ be a random $\pm 1$ walk on the $x$-axis of $2n$ steps, and $A$ be a random set of $t$ odd moments from $\{1, \ldots, 2n - 1\}$, then put $X_{n, t}$ as the number of elements of $A$ before the last visit of $0$ in $W$. Then the distribution of $X_{n, t}$ is independent of $n$, as adjusted RHS$(\heartsuit)$ is $\Pr(X_{n + p + q, p + q}) = p$. | |
| Mar 10, 2023 at 21:26 | history | edited | Mikhail Tikhomirov | CC BY-SA 4.0 |
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| Mar 10, 2023 at 21:20 | history | edited | Mikhail Tikhomirov | CC BY-SA 4.0 |
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| Mar 10, 2023 at 20:57 | history | answered | Mikhail Tikhomirov | CC BY-SA 4.0 |