Journal of Samara State Technical University, Ser. Physical and Mathematical SciencesJournal of Samara State Technical University, Ser. Physical and Mathematical Sciences1991-86152310-7081Samara State Technical University2076710.14498/vsgtu1333Research ArticleOn the one property of the free components concerning to the sum of equal powersNikonovAlexander I(Dr. Techn. Sci.; nikonovai@mail.ru), Professor, Dept. of Electronic Systems and Information Securitynikonovai@mail.ruSamara State Technical University1509201418316116818022020Copyright © 2014, Samara State Technical University2014The given paper contains the proof of that the number of combinatorial arrangements coincides with free components of the sums of equal powers with the natural bases and parameters in the presence of the simple equality connecting elements of these arrangements. In the proof the modified exposition of the components participating in formation of the sum of equal powers is used. This exposition becomes simpler and led to an aspect of product of binomial factors. Other variants of construction of corresponding product of binomial factors do not exist here. The received proof allows both to represent number of arrangements in the form of product, and to apply at this representation summation elements. Thus, the number of arrangements supposes characteristic expression not only in the form of product of its elements.sum of equal powersfree componentsnumber of arrangementsbinomial factorsсумма одинаковых степенейсвободные компонентычисло размещенийбиномиальные коэффициенты[Beery J. Sums of Powers of Positive Integers: Loci (July 2010), 2010. doi: 10.4169/loci003284.][Oral H. K., Unal H. Extending al-Karaji’s Work on Sums of Odd Powers of Integers: Loci (August 2011), 2011. doi: 10.4169/loci003725.][Wang X., Yang S. On solving equations of algebraic sum of equal powers // Science in China Series A: Mathematics, 2006. vol. 49, no. 9. pp. 1153-1157. doi: 10.1007/s11425-006-1153-y.][De Koninck J.-M., Luca F. Integers divisible by sums of powers of their prime factors // Journal of Number Theory, 2008. vol. 128, no. 3. pp. 557-563. doi: 10.1016/j.jnt.2007.01.010.][Torabi-Dashti M. Faulhaber’s Triangle // The College Mathematics Journal, 2011. vol. 42, no. 2. pp. 96-97. doi: 10.4169/college.math.j.42.2.096.][Almismari N. A new method to express sums of power of integers as a polynomial equation: viXra:1211.0102, 2012. 9 pp.][Guo S., Shen Y. On Sums of Powers of Odd Integers // Journal of Mathematical Research with Applications, 2013. vol. 33, no. 6. pp. 666-672. doi: 10.3770/j.issn:2095-2651.2013.06.003.][Suprijanto D., Rusliansyah. Observation on sums of powers of integers divisible by four // Applied Mathematical Sciences, 2014. vol. 8, no. 45. pp. 2219-2226. doi: 10.12988/ams.2014.4140.][Cereceda J. L. A determinant formula for sums of powers of integers // International Mathematical Forum, 2014. vol. 9, no. 17. pp. 785-795. doi: 10.12988/imf.2014.4461.][Никонов А. И. Об одном свойстве взвешенных сумм одинаковых степеней как матричных произведений // Вестн. Сам. гос. техн. ун-та. Сер. Физ.-мат. науки, 2010. № 5(21). С. 313-317. doi: 10.14498/vsgtu816.][Никонов А. И. Модифицированное описание компонентов, образующих сумму взвешенных одинаковых степеней // Вестн. Сам. гос. техн. ун-та. Сер. Физ.-мат. науки, 2012. № 1(26). С. 223-232. doi: 10.14498/vsgtu1016.][Никонов А. И. Комбинаторное представление суммы взвешенных одинаковых степеней членов арифметической прогрессии // Вестн. Сам. гос. техн. ун-та. Сер. Физ.-мат. науки, 2013. № 4(33). С. 184-191. doi: 10.14498/vsgtu1288.][Haggarty R. Discrete mathematics for computing. Harlow: Addison-Wesley, 2002. 235+xii pp.][Strang G. Linear Algebra and its Applications. 2nd ed. New York, San Francisco, London: Academic Press, 1980. 414+xi pp. doi: 10.1016/B978-0-12-673660-1.50001-0][Riordan J. An introduction to combinatorial analysis. Princeton, New Jersey: Princeton University Press, 1980. 244+xii pp.]