It is shown that Floyd's subset sampler admits an exact round-local factorization of an additional ordering coordinate that is absent from the returned set and gives a finite counterexample showing that analogous immediate rank recycling in a partial Fisher-Yates array is invalid.
Abstract
A uniformly random $m$-subset of $[n]=\{0,\ldots,n-1\}$ has entropy $\log_2\binom{n}{m}$. Standard without-replacement procedures often expose an additional ordering coordinate that is absent from the returned set. We show that Floyd's subset sampler admits an exact round-local factorization of this coordinate. In round $r$, let $S$ be an $(r-1)$-subset of $[j]$, let $T\sim\operatorname{Unif}([j+1])$, and let $S'$ be the result of Floyd's transition. If $D$ is the zero-based rank of the original draw $T$ in $S'$, then $(S,T)\leftrightarrow(S',D)$ is a bijection between $\binom{[j]}{r-1}\times[j+1]$ and $\binom{[j+1]}{r}\times[r]$. Consequently, $S'$ and $D$ are independent and uniform on their respective spaces. The digit $D$ can therefore be merged immediately into a residual uniform random state; an induction shows that the partial subset remains independent of that state after every round. For $k=\min(m,n-m)$, the sampling phase uses $O(k\log k)$ time and $O(k)$ auxiliary space with an order-statistic tree; explicitly materializing a complement incurs the unavoidable output cost. The combinatorial layer avoids binomial-coefficient arithmetic and recovers the complete $k!$ state-space factor exactly. We also give a finite counterexample showing that analogous immediate rank recycling in a partial Fisher-Yates array is invalid because the unselected suffix retains a correlated ordering. A 64-bit Rust implementation is checked by exhaustive state-space enumeration for all $n\leq 8$ and by an entropy-accounting trace for choosing $20{,}000$ of $30{,}000$ items. We make no claim of runtime superiority over existing subset samplers.
Let $C\subseteq\F_2^m$ be a binary linear code and let $[m]=L\sqcup R$ be a bipartition of its coordinates. The \emph{conditional decoding matrix} of $C$ at this cut is the matrix $W$ indexed by $\F_2^{L}\times\F_2^{R}$ whose entry $W(x_L,x_R)$ is the coset-leader weight $d\bigl((x_L,x_R),C\bigr)$, the minimum Hamming distance from the word $(x_L,x_R)$ to the code. We prove that the min-plus factorization rank (Barvinok rank) of $W$, and likewise its tropical rank, equal $2^{s}$ exactly, where $s=\dim C-\dim C_L-\dim C_R$ is the classical state complexity of the minimal trellis of $C$ at the cut. The upper bound is a two-party reading of Viterbi decoding on the minimal trellis; the contribution is the matching lower bound, which holds against arbitrary min-plus factorizations rather than only sequential trellis realizations, and is obtained from an explicit $2^{s}\times 2^{s}$ tropically nonsingular submatrix built from a transversal of codewords. Specializing $C$ to the cut space of a graph identifies $W$ with the conditional ground-state energy of Ising signings (the frustration index), and yields natural graph families whose conditional matrices have min-plus rank exponential in the number of vertices; for these families we also record the contrasting local statement that all bounded-radius views of a signing are switching-trivial, so the exponential rank is carried entirely by non-local structure. We note explicitly that this rank measures representational incompressibility, not computational hardness: planar families attain the same exponential rank while their ground states are computable in polynomial time.
This work refine the existing parameter estimation guarantees under the fatness assumption, improving the prior sample complexity to $O( \log n / \epsilon^2)$ for $\ell_\infty$-recovery, matching the untruncated minimax rate.
Rohan Chauhan, Ioannis Panageas· arXiv.org· 0 citations
For integers $n\ge s\ge2$, let $e(n,s)$ be the maximum size of a family $\mathcal F\subseteq2^{[n]}$ with no $s$ pairwise disjoint members. The problem of determining $e(n,s)$, now called the Erd\H{o}s--Kleitman problem, is closely related to the well-known Erd\H{o}s matching problem. Frankl and Kupavskii posed a meta-conjecture predicting that the maximum is always attained by a weighted family. Fix $m\ge3$, write $n=ms+c$ with $0\le c<s$, and set $\ell=s-c$. For $0\le k\le m$, let $a_k=ms-kc-1$. For $A\in\binom{[n]}{a_k}$, define \[ \mathcal H^k(m,s,\ell;A):= \{F\subseteq[n]: k|F|+|F\cap A|\ge m(k+1)\}. \] This defines a unified class of weighted families with matching number less than $s$. Among these families, $\mathcal H^0$, $\mathcal H^1$, and $\mathcal H^m$ were previously known to be extremal in different ranges of $c$. We show that for $1\le k\le m-1$, all families $\mathcal H^k$ are uniquely extremal in some ranges of $c$. More precisely, we prove that for every $m\ge3$ and every $1\le k\le m-1$, there exist constants $\alpha=\alpha(m,k)>0$, $\beta=\beta(m,k)>0$ and an integer $s_0=s_0(m,k)$ such that, for all integers $s\ge s_0$ and all integers $c$ with $0\le c<s$, the only extremal families for $e(n,s)$ are the families $\mathcal H^k(m,s,\ell;A)$ with $A\in\binom{[n]}{a_k}$ whenever $\beta s^{(k-1)/k}\le c\le \alpha s^{k/(k+1)}$. In particular, this result determines an infinite number of new extremal families for the Erd\H{o}s--Kleitman problem and verifies the Frankl--Kupavskii meta-conjecture in these ranges. This also provides a quantitative extension of the result of Kupavskii and Sokolov on the extremality of $\mathcal H^1$.
Let $P$ be a finite nonempty poset with $n$ elements, let $f:P\to\{1,\ldots,n\}$ be a uniformly random order-preserving bijection, and put $h_P(x)=\mathbb{E}[f(x)]$. Aires and Kahn (2025) introduced $\operatorname{gap}(P)$ as the largest difference between consecutive values in the ordered list consisting of $0$, $n+1$, and all the expected ranks $h_P(x)$. Write ${w}(P)$ for the largest size of a pairwise incomparable subset. We prove three results. First, we prove a weighted strengthening of an ideal inequality conjectured by Kahn and obtain the explicit gap-width bound $\operatorname{gap}(P)\le 2 {w}(P)-1$. Second, for every $L>0$ we construct a width-two poset such that the expected-rank list of every maximal chain has a gap of at least $L$, with $0$ and $|P|+1$ added as endpoints. Finally, for every $r\in\mathbb{N}$, we construct a poset $P_r$ for which the relative order induced on every nonempty selected set $X$ has base-two entropy below $3|X|$, while $\operatorname{gap}(P_r)\ge(3/2)^r$. Thus the gap can be arbitrarily large while the induced order on every selected set has relatively small entropy. The key ideas behind all three results were found by ChatGPT 5.6 Sol.
For primes $n\equiv1\pmod{24}$ we study how deep an explicit, factorization-free search for a decomposition of $4/n$ into three unit fractions has to go. Write $E_2(N;J)$ for the set of such primes $n\le N$ at which no witness of depth at most $J$ exists, in the sense of the two-parameter criterion of Theorem 3.9 with coprime parameters $u,a\le J$. We prove that for every fixed $J$ $$|E_2(N;J)| \ll_J \frac{N}{(\log N)^{1+\mathfrak{A}(J)/2}},$$ the exponent being the exact dimension of the covering on which the proof rests. The proof replaces the subgroup generated by the prime factors, an approach that breaks down as soon as $(\mathbb{Z}/4m)^{\times}$ has exponent greater than $2$, by a fixed-point-free involution, and is unconditional at every $J$. Second, we exhibit an unconditional obstruction. At the shift $c=7$ there are $\asymp N(\log N)^{-3/2}$ primes $n\le N$, $n\equiv1\pmod{24}$, at which both branches of the divisor criterion fail. The representation of $K_7=(n+7)/4$ by the principal form of discriminant $-7$ has to be primitive, so that the relevant input is the primitive-representation theorem of Fuchs, Hsu, Rickards, Schindler and Stange [25] rather than the classical results of Iwaniec; a fixed shift therefore cannot leave a finite residual set. Third, a factorization-free procedure decides the conjecture for all primes of an interval $[N,2N]$. Its Type II pass costs $\mathcal{O}(N(\log N)^{3})$ while its Type I pass costs $\Theta(N^{2})$, which locates the whole quadratic cost in the extraction of the divisors of $4u^{2}d+1$ and exhibits an asymmetry between the two halves of the Type I/Type II dichotomy. The conjecture itself remains open.
We show that some classical identities valid for ordinary partitions have precise analogues for $n$-color partitions, that is partitions in which a part of size $n\geq 1$ can occur in colors $1, 2, \ldots, n$. For $r \ge 2$ and $j \ge 0$, we write $\mathcal{O}_{j,r}(m)$ and $\mathcal{D}_{j,r}(m)$ for the sets of $n$-color partitions of $m$ with, respectively, exactly $j$ different parts whose size and color are divisible by $r$, and exactly $j$ different parts occurring at least $r$ times. We prove an $n$-color version of Franklin's theorem, $|\mathcal{O}_{j,r}(m)| = |\mathcal{D}_{j,r}(m)|$, along with two Beck-type identities. We give both analytic and combinatorial proofs for all theorems.
C. Ballantine, R. Tauraso· 0 citations
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