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Jul 11 |
comment |
Which subgroup order of the symmetric group is the most frequent?
@EricWofsey: It seems like a difficult question actually, see the top answer here: math.stackexchange.com/questions/76176/… |
Jul 9 |
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Another name for coin-flipping polynomials
They are homogenous polynomials for each $n$ in variables $x,y=(1-x)$ I'm not sure what else you can say without making some kind of restrictions. |
Jul 9 |
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Another name for coin-flipping polynomials
A general singleton has probability $x^k(1-x)^{n-k}$ in the coin space. Any event is thus $\sum_{k\in E}x^k(1-x)^{n-k}$, where $E$ is an index multiset set of the event, which is the form of the most general coin polynomial. |
Jul 5 |
comment |
Distribution of trivial subset sums
If you add $L+1$ to the interval, then you're looking at the number of ways to sum to $L+1$ from $[1,L]$. You necessarily need all your chosen numbers to be less than $L-1$, which occurs with probability $\binom{L-1}{n}$, after which you are evaluating $p(L-1)$: the number of partitions of $L-1$ with distinct parts, which equals the number of partitions with odd parts by Euler's theorem. I'm guessing the growth rate is known via circle methods or other analytic combinatorics, probably something like $O(\exp(c\sqrt{n})/n)$. |
Jun 25 |
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Logarithmic integral, $π(x)$ and $x/(\ln x)$
Not yet. See the discussion here: en.wikipedia.org/wiki/Skewes'_number#More_recent_estimates |
Jun 25 |
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Extrapolation between longest increasing and longest alternating subsequences
A trivial observation: there's some strange parity here. The Tracy widom distribution is biased toward one side, whereas the Gaussian is symmetric about the mean. This is evident here since for small $m$, we basically have reflection symmetry. So I would expect the transition to occur when $m\gg n/2$. |
Jun 23 |
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Semicircle law universality elsewhere
thanks for all your answers! |
Jun 23 |
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Bounds on the probability of k-of-n events in terms of bounds on single and pairwise probabilities
It sounds like you could really use something akin to positive/negative association such as $P(A_i\cap A_j)\leq \geq P(A_i)P(A_j)$. Do you happen to know if these hold on some large subset of index pairs? |
Jun 22 |
revised |
Semicircle law universality elsewhere
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Jun 22 |
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Why is this distribution exponential?
Try working this out with just two points. |
Jun 22 |
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Why is this distribution exponential?
Basically by rescaling the interval as a function of $n$ the waiting time between successive points becomes a poisson process. This is because the probability of seeing the next point is proportional to the interval you are looking at. |
Jun 19 |
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Semicircle law universality elsewhere
@jon bannon: I'm not intimately familiar with it but I basically lumped it with random matrix theory. If this is wrong of me I would love to see a note on this. |
Jun 19 |
revised |
Semicircle law universality elsewhere
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Jun 19 |
asked | Semicircle law universality elsewhere |
Jun 18 |
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Determinant Evaluation
@SteveHuntsman: I have, for example "A determinental evaluation and some enumeration results for plane partitions": citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.23.3644 It seems like this essentially counts plane-partitions bounded by $c$. In the paper it's mentioned that there's no known explicit formula in the general case but I'm wondering if these have been considered elsewhere? In the linked paper, Theorem 5 has something similar but by a (intended) miracle, the calculation goes through. |
Jun 18 |
revised |
Determinant Evaluation
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Jun 17 |
revised |
Determinant Evaluation
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Jun 17 |
asked | Determinant Evaluation |
May 19 |
asked | Character sums over a fixed subset of skew tableaux |
May 18 |
comment |
Young Tableau Box Correlations
@oferzeitouni: I'm familiar with the mentioned lozenge tiling results but unfortunately I'm not sure if they apply here. Specifically, lozenge tilings use Plancherel measure whereas this problem is for uniform measure on a fixed tableau. As well, as far as I remember, schur functions and macdonald polynomials index lozenge tile locations and don't really rely on exact single box distributions. The only connection I see to Plancherel measure is that the resulting square limit curves are in some sense deformed Logan-Shepp curves. |