D) number of even components. ThenMathematics 2021, 9,7 ofBe (n) – Bo (n) =1if n = m(4m 1), m 0; otherwise.Note that the creating function for the sequence B(0), B(1), B(two), . . . is( q2 ; q2)Hence,q2(135…2n-1) (-q2n2 ; q2) (q; q2)n n =n =( Be (n) – Bo (n))qn = (q2 ; q2) q2(135…2n-1) 2n2 two (q ; q) (q; q2)n n =q2n (q2 ; q2) = two 2 two n=0 ( q; q)n ( q ; q)n= ( q2 ; q2)q2n (q; q)2n n == =and the outcome follows.n =(1 q8n-3)(1 q8n-5)(1 – q8n) by (two)q4n2 n,n=-Corollary 2. For all n 0, B(n) is odd if and only if n = m(4m 1) for some integer m 0. Ultimately, contemplate the partition function; (n): the number of partitions of n in which even parts are distinct or if an even aspect is repeated, it’s the smallest and occurs specifically twice and all other even parts are distinct. Let e (n) (resp. o (n)) denote the number of (n)-partitions with an even (resp. odd) number of distinct even components. Then, the following identity follows: Theorem 7. For all non-negative integers n, we’ve got e (n) – o (n) = exactly where e (0) – o (0) := 1. Proof. Note that 1, 0, if n = 3m, 3m 1, m 0; otherwisen =(n)qn ==(-q2 ; q2) q2n2n (-q2n2 ; q2) (q2n1 ; q2)-1 (q; q) n =n =q2n2n (-q2n2 ; q2) (q2n1 ; q2)-Mathematics 2021, 9,8 ofso thatn =(e (n) – o (n))qn = q2n2n (q2n2 ; q2) (q2n1 ; q2)-= =n =0 ( q2 ; q2)(q; q2) (q; q2)n =q4n( q2 ; q2)n(q; q2)nqn( q2 ; q2)4 two ( q ; q) n =(q; q2)( q2 ; q2)n( q4 ; q2) nby (3), a = c = 0, b = q, t = q4 .= (1 – q2) = (1 – q2)qn ( q4 ; q2) n ( q2 ; q2) n n =(1 – q2n2)qn 1 – q2 n == = =n =(1 – q2n2)qn qn – q3nn =0 n =n =0 n =q3n q3n1 .Instance 2. Take into account n = 8. The (8)-partitions are:(8), (7, 1), (six, two), (5, 3), (five, 1, 1, 1), (4, 4), (four, two, 1, 1), (three, 3, 1, 1), (3, 1, 1, 1, 1, 1), (6, 1, 1), (five, two, 1), (4, three, 1), (4, two, 2), (4, 1, 1, 1, 1), (three, three, two), (3, 2, 1, 1, 1), (2, 1, 1, 1, 1, 1, 1), (1, 1, 1, 1, 1, 1, 1, 1).The e (8)-partitions are:(7, 1), (6, 2), (five, 3), (five, 1, 1, 1), (4, four), (4, 2, 1, 1), (3, 3, 1, 1), (three, 1, 1, 1, 1, 1), (1, 1, 1, 1, 1, 1, 1, 1),and o (8)-partitions are:(eight), (6, 1, 1), (5, 2, 1), (four, 3, 1), (4, 2, two), (4, 1, 1, 1, 1), (3, 3, two), (3, two, 1, 1, 1), (two, 1, 1, 1, 1, 1, 1),Indeed e (eight) – o (8) = 0. The above theorem is usually used to establish the parity of (n). We write down this as a consequence in the corollary below. Corollary three. For all n 0, (n) is odd if and only if n 0, 1 (mod 3). four. Conclusions A lot as we couldn’t generalize Theorem two by means of generating functions, we supplied a generalization via a bijective construction. Several partition functions that happen to be related to the theorem were studied. Our L-?Leucyl-?L-?alanine MedChemExpress investigation incorporated deriving parity formulas and establishing new partition identities. Of unique interest was Theorem 7 whose combinatorial proof we seek.Mathematics 2021, 9,9 ofAuthor Contributions: Funding acquisition, A.M.A.; Investigation, A.M.A. and D.N.; Methodology, D.N.; Supervision, D.N.; Validation, A.M.A.; Writing riginal draft, D.N.; Writing eview diting, A.M.A. All authors have read and agreed for the published version of the manuscript. Funding: The authors extend their appreciation for the Deanship of Scientific Investigation at University of Tabuk for funding this function by means of Analysis Group no. RGP-0147-1442. Institutional Assessment Board Nimbolide Inhibitor Statement: Not applicable. Informed Consent Statement: Not applicable. Information Availability Statement: Not applicable. Conflicts of Interest: The authors declare that they have no conflict of interest.mathematicsArticleTowards Optimal Supercomputer Energy Consumption Forecasting MethodJiTom al.