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Invariants in partition classes
AIMS Mathematics ( IF 2.2 ) Pub Date : 2020-08-05 , DOI: 10.3934/math.2020401
Aleksa Srdanov ,

With $ p\left(n,k\right) $ denote the numerical value of the number of partitions of the natural number $ n $ on exactly $ k $ parts. Form an arithmetic progression of $ k $ natural numbers with an arbitrary first value $ x_1=p\left(j,k\right)$, and the difference $ d=m \cdot LCM\left(1,2,\dots,k\right) $, where $ j$ and $ m $ an arbitrary natural numbers. Calculate all the values of $ \left\{p\left(x_i,k\right)\right\}_{i=1,2, \dots,k} $ and make the alternating sum with the appropriate binomial coefficients $ \sum_{i=0}^{k-1}\left(-1\right)^i \binom{k-1}{i}p\left(j+i\cdot d,k\right). $ The last sum has a constant value equal to $ \left(-1\right)^{k-1}\frac{d^{k-1}}{k!} $, regardless of the first selected member $ x_1 $ of the arithmetic progression. We call this sum the first partition invariant, and it exists in all classes. In addition to these values there are a whole number of other invariant values, but they exist only in some classes, and so forth.

中文翻译:

分区类中的不变量

其中$ p \ left(n,k \ right)$表示正好$ k $部分上自然数$ n $的分区数的数值。形成具有任意第一个值$ x_1 = p \ left(j,k \ right)$的$ k $自然数的算术级数,并且差$ d = m \ cdot LCM \ left(1,2,\ dots, k \ right)$,其中$ j $和$ m $是任意自然数。计算$ \ left \ {p \ left(x_i,k \ right)\ right \} _ {i = 1,2,\ dots,k} $的所有值,并用适当的二项式系数$ \进行交替求和。 sum_ {i = 0} ^ {k-1} \ left(-1 \ right)^ i \ binom {k-1} {i} p \ left(j + i \ cdot d,k \ right)。$最后一个总和的常量值等于$ \ left(-1 \ right)^ {k-1} \ frac {d ^ {k-1}} {k!} $,而不考虑第一个选定的成员$ x_1算术级数的$。我们将此总和称为第一个分区不变式,它存在于所有类中。
更新日期:2020-08-05
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