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Combinatorics
Published in Paul L. Goethals, Natalie M. Scala, Daniel T. Bennett, Mathematics in Cyber Research, 2022
To count involutions, we split them into cases based on the number of fixed points. If a permutation of length 2k has no fixed points, the remaining entries must be grouped in pairs. We have 2k−1 options for the first image, 2k−3 options for the next, and so on, leading to (2k−1)(2k−3)⋯3·1. This expression is called the double factorial of 2k−1, and is denoted (2k−1)!!. Summing over all possible numbers of fixed points proves the following proposition.
Existence Theorems and Special Functions
Published in L.M.B.C. Campos, Singular Differential Equations and Special Functions, 2019
where: (i) the double factorial (9.365a) is used in (9.365c; 9.366c); (ii) the recurrence relation (9.365b) is applied n – 1 times leading to (9.365c) because c01 is not zero for (9.363a) in (9.362b); (iii) the recurrence relation (9.366b) is applied n–2 times leading to (9.366c) to end at c11/2, short of c01/2=0 for (9.364a) in (9.362b).
Integral Equations in Scattering Problems
Published in Edward F. Kuester, David C. Chang, Electromagnetic Boundary Problems, 2015
Edward F. Kuester, David C. Chang
where the double factorial notation signifies () (2n−1)!!=1⋅3⋅5⋯⋯(2n−1)=(2n)!2nn!
New oil and forage producing plant species evaluated on phosphorus doses and row spacing
Published in Biofuels, 2023
Rodrigo Gomes Branquinho, José Milton Alves, Madalena Arantes da Silva, Karine Feliciano Barbosa, Edson Luiz Travagin
Seeds were collected from the local flora, and sown in trays for seedling production on 02/15/2017. Before sowing, germination and vigor tests were performed. Seedlings were transplanted in experimental plots measuring 3 × 3 m. The fertilization with triple superphosphate, as well as the spacing between rows of plants, varied according to the treatment used. The treatments consisted of four concentrations of P2O5 (0, 40, 80 and 160 kg ha−1) and two spacings between rows (0.25 and 0.50 m), combined in a double factorial scheme 4 × 2. The treatments were distributed to the plots following a randomized complete block design, with four replications of the base experiment. The plant population was 120 and 240 thousand plants per hectare, for the spacing of 0.25 and 0.50 m, respectively.
Experimental validation of unique combination numbers for third- and fourth-order neutron correlation factors of zero-power reactor noise
Published in Journal of Nuclear Science and Technology, 2019
Tomohiro Endo, Akio Yamamoto, Masao Yamanaka, Cheol Ho Pyeon
As shown in Figure 1, the unique combination numbers of near the critical state can be inferred by the heuristic enumeration method for . Based on the heuristic method for the complete (unordered) binary tree, as shown in Figure 1, the unique combination numbers can be deduced using the double factorial [36]:
Theoretical Derivation of a Unique Combination Number Hidden in the Higher-Order Neutron Correlation Factors Using the Pál-Bell Equation
Published in Nuclear Science and Engineering, 2023
Tomohiro Endo, Fuga Nishioka, Akio Yamamoto, Kenichi Watanabe, Cheol Ho Pyeon
Considering that the saturation value of the second-order correlation factor converges to , we can finally find that the ratio of is equal to the double factorial for any system in the critical state: