Tu-Deng问题中的循环删除刚性与Macaulay阴影

中文摘要 AI 辅助

我们确定了Tu-Deng界$|S_{t,k}|\le 2^{k-1}$中的所有等号情形。若$t$的$k$位循环字有$R$个1、$Z$个0，且循环1-间隙长度为$g_1,\ldots,g_Z$，则等号成立当且仅当对每个$i$都有$g_i\ge Z-1$。这解决了Flori、Randriambololona、Cohen和Mesnager的猜想3.20，且我们还枚举了所有等号参数。对$R\ge Z$，我们确定了尖锐的第一稳定性间隙和所有极端字；对$R<Z$，我们得到了亏空的精确量化和显式的依赖游程的下界。证明是结构性的：显式矩阵共轭识别了Tu-Deng猜想的两个近期完整证明中的辅助枚举器。随后我们为所有系数开发了一个带根的粗化模型，证明了单侧删除刚性和一个精确的Macaulay通量恒等式，并从最高循环层级的有界单纯形导出了Macaulay-Möbius公式。

英文摘要

Let $N=2^k-1$, and let $S_{t,k}$ consist of the pairs $0\le a,b<N$ such that $a+b\equiv t\pmod N$ and $\wt(a)+\wt(b)<k$. The Tu--Deng bound $|S_{t,k}|\le 2^{k-1}$ has recently been proved. In this paper, we give a combinatorial proof of the equality criterion. If the $k$-bit cyclic word of $t$ has $Z$ zeros and $g_1,\ldots,g_Z$ are the numbers of ones between successive zeros, then $$|S_{t,k}|=2^{k-1} \quad\Longleftrightarrow\quad g_i\ge Z-1\quad(1\le i\le Z).$$ Moreover, we recover the resulting enumeration of the equality parameters. Beyond equality, if $R\ge Z\ge2$, where $R$ is the number of ones, then every nonequality parameter satisfies $$2^{k-1}-|S_{t,k}|\ge 2^{R-Z+1},$$ and we classify all cases in which this bound is attained. For $R<Z$ we obtain a congruence for $|S_{t,k}|$ and a lower bound for the deficit in terms of the number of cyclic runs of ones.

发表机构

• China West Normal University（西华师范大学）

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17 pages