整数分拆的迭代不同绝对差
Iterated Distinct Absolute Differences of Integer Compositions
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中文总结 AI 辅助
该研究确定了具有指定深度的整数分拆对应的最小正整数，推导了其公式，证明了≥该值的整数均有对应深度分拆，还对达最小a(n)的分拆进行分类计数并给出公式。
中文摘要 AI 辅助
从一个整数分拆出发，构造其连续绝对差，要求这些差非零且两两不同，随后可对这些差进行任意排列后重复该操作。分拆的深度是其能进行的最大连续迭代次数。我们确定了能达到任意指定深度的分拆所对应的最小正整数：若a(n)是具有深度n的分拆的最小正整数，则a(n)=n+1+⌈n(n+1)/4⌉+⌊n/2⌋。我们还证明了每个至少为a(n)的整数k都具有深度至少为n的分拆，因此若d(k)表示整数k的分拆的最大深度，则d(k)=max{n≥0:a(n)≤k}。最后，我们对所有达到最小a(n)的分拆进行分类和计数，其数量由四个阶乘公式给出，具体取决于n模4的结果。
英文摘要
Starting from an integer composition, form its consecutive absolute differences, provided that they are nonzero and pairwise distinct, and then permute these differences arbitrarily before repeating the operation. The depth of the composition is the maximum possible number of successive iterations. We determine the least positive integer admitting a composition of any prescribed depth. If a(n) is the least positive integer having a composition of depth n, then $a(n)=n+1+\lceil n(n+1)/4\rceil+\lfloor n/2\rfloor$. We also prove that every integer k at least a(n) has a composition of depth at least n. Consequently, if d(k) denotes the maximum depth of a composition of k, then $d(k)=\max\{n\geq 0:a(n)\leq k\}$. Finally, we classify and enumerate all compositions attaining the minimum a(n). Their number is given by four factorial formulas according to n modulo 4.
