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19h
comment Analysis of Nim-Like Game?
bof: I wouldn't say that. First, some Nim-type games are equivalent to Nim with one pile of size up to $n$. Second, there are infinitely many moves possible when you have a heap of size greater than $1$, so a priori there could be infinite ordinal nimbers necessary to describe the game.
20h
comment Analysis of Nim-Like Game?
Again, the analysis of this game is an exercise. The only way this might be on-topic is as a reference request. Perhaps people have looked at families of games on partitions generalizing this one.
20h
comment Analysis of Nim-Like Game?
@bof: That looks like a mistake, but you can move from $[1,2,3]$ to $[1,2,2,2]$ which is equivalent to $[1,2]$.
20h
comment Analysis of Nim-Like Game?
By the way, this is equivalent to Nim with one pile, though it takes a little effort to see the correspondence. The winning positions for the second player have an even number of heaps of each size. It's easy to find the nimber of a position base $2$.
23h
comment Analysis of Nim-Like Game?
I don't think the analysis is research-level, but it might be ok as a reference request.
1d
awarded  co.combinatorics
2d
comment Regarding left-to-right minima
Second, I used the union bound. For the max of $l_1, ..., l_n$ to be at least $x$, at least one of them must be, so the probability that the max is at least $x$ is at most the sum of the probabilities that each $l_i$ is at least $x$.
2d
comment Regarding left-to-right minima
I used two steps there. First, there is a standard trick that comes from rewriting a non-negative integer-valued random variable $M$ as a sum of indicators of whether the random variable is at least n. $M = Z_1 + Z_2 + ...$ so $E[M] = E[Z_1 + Z_2 + ...] = E[Z_1] + E[Z_2] + ... = P(Z_1=1) + P(Z_2=1) + ...$. Let $h= \lfloor 1+\epsilon H_n \rfloor, M = \max(h,\max(l_1,...,l_n)).$ Then $M = h + 1_{\{M \ge h+1\}} + 1_{\{M \ge h+2\}} + ...$ so $E[\max(l_1,...,l_n)] \le E[M] = h + P(M \ge h+1) + P(M \ge h+2) + ...$.
2d
revised Regarding left-to-right minima
Corrected misplaced epsilon^2 in one formula. Made trivial changes.
2d
answered Regarding left-to-right minima
2d
comment Regarding left-to-right minima
By $\rho_{i...n}$, you mean the suffix of the permutation, $j \mapsto \rho(j)$ for $i \le j \le n$?
Apr
23
comment Size of smallest multiple $\sum \epsilon_i p^i$ of $n$ with digits $\epsilon_i\in \{0,1\}$ for $p$ a prime
Which part doesn't make sense for non-prime $p$?
Apr
23
comment Size of smallest multiple $\sum \epsilon_i p^i$ of $n$ with digits $\epsilon_i\in \{0,1\}$ for $p$ a prime
Coincidentally, I just answered a different but related question here: programmers.stackexchange.com/questions/272517/…
Apr
22
comment Trapped Billiard trajectories on non-convex billiard tables
The paper you cite has the assumption that the curvature doesn't vanish anywhere. If the curvature at a point doesn't vanish, and the third derivative is bounded in a neighborhood of the point, then no trajectory can get stuck at the point. Global failure of convexity isn't important. So, what remains is to consider the cases where the curvature vanishes.
Apr
21
comment First collision time of $n$ random walkers on a cycle
That definition of collision isn't a function of the resulting position, though.
Apr
21
comment Determinant of a checkerboard Hankel matrix with Catalan numbers
What are the first few values? Anything in the Online Encyclopedia of Integer Sequences?
Apr
21
revised Maximum occupancy balls in bins with limited independence
Used convex substitute for binomial.
Apr
19
comment Raising coefficients of a power series to some power
David Handelman pointed out that this is the Hadamard product of P with itself. Hadamard proved some results that might be useful, perhaps in J. Hadamard, Th´eoreme sur les s´eries entieres, Acta. Math. 22 (1899) 55–63. I haven't tracked down that reference yet, though papers that cite it mention some theorems that might be useful.
Apr
18
answered Maximum occupancy balls in bins with limited independence
Apr
18
answered Decomposition of a cross-polytope into simplices