蝴蝶序列：具有不同部分的整数分拆数的第二差序列，它的五边形数结构，它的组合恒等式和分圆多项式1-x和1+x+x^2
The butterfly sequence: the second difference sequence of the numbers of integer partitions with distinct parts, its pentagonal number structure, its combinatorial identities and the cyclotomic polynomials 1-x and 1+x+x^2
摘要：
在作者以前关于扭相对顶点算子代数和扭相对顶点算子模的雅可比恒等式以及仿射李代数的生成函数恒等式的基础上，我们解释了不同部分整数分拆个数序列的二阶差序列（严格划分）作为n的严格划分的序列，至少有三个部分，三个最大的部分是连续的，而最小的部分至少是两个。蝴蝶这个名字既描述了序列的解释，也描述了正整数m的两个最大部分连续的严格划分的集合与m+1的同类严格划分的子集之间的基本双射。利用分圆多项式1-x和1+x+x^2，计算了蝴蝶序列及其相关序列的生成函数恒等式，它们既是无穷乘积，又是按相应分块的部分数过滤的级数，并且我们看到，正整数n的奇数部分大于或等于5的分块数是蝴蝶序列的三个连续项之和.我们还确定了一个微妙的合并和分裂建设的蝴蝶序列作为一个序列的一些分区的奇数部分大于或等于3，我们提供了一个相关的详细解释的蝴蝶序列作为一个序列，我们定义为广义五边形，五边形与多米诺骨牌，和非五边形蝴蝶分区。最后，欧拉的五边形数定理和Jacobi三重积的稍微不同的特殊化导致递归算法来计算蝴蝶序列和使用五边形数序列和三角幂级数的相关序列。
英文摘要：
Based on the author's previous work on the Jacobi identity for twisted relative vertex operator algebras and modules and on the generating function identities for affine Lie algebras, we interpret the second difference sequence of the sequence of the number of integer partitions with distinct parts (the strict partitions) as the sequence of the strict partitions of n with at least three parts, the three largest parts consecutive, and the smallest part at least two. The name butterfly describes both the sequence's interpretation and the underlying bijection between the set of strict partitions of positive integers m with the two largest parts consecutive, and a subset of the same kind of strict partitions of m+1. Using the cyclotomic polynomials 1-x and 1+x+x^2, we compute generating function identities o the butterfly sequence and related sequences both as infinite products and as series filtered by the number of parts of the corresponding partitions, and we see that the number of partitions of positive integers n with odd parts greater than or equal to 5 is the sum of three consecutive terms of the butterfly sequence. We also determine a subtler merging and splitting construction of the butterfly sequence as a sequence of some of the partitions with odd parts larger or equal to 3, and we offer a related detailed interpretation of the butterfly sequence as a sequence of what we define as generalized pentagonal, pentagonal with domino, and non-pentagonal butterfly partitions. Finally, Euler's Pentagonal Number Theorem and a slightly different specialization of the Jacobi triple product lead to recursive algorithms to compute the butterfly sequence and related sequences using pentagonal number sequences and the series of triangular powers.
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