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Binomial Inversion Equation (3) is an example of something called a binomial transform, and I will refer to identities such as (1) as “orthogonality relationships”, borrowing some terminology from linear algebra
combinatorics - Combinatorial interpretation of Binomial Inversion . . . Is there a nice combinatorial interpretation of this phenomena? Nice applications? Are there any more famous cool inversions (I know of Möbius inversion, binomial inversion, and the discrete derivative inversion ai → ai+1 −ai) a i → a i + 1 − a i)?
Binomial transform - Wikipedia The binomial transform can be written in terms of binomial convolution Let and for all Then The formula can be interpreted as a Möbius inversion type formula since is the inverse of under the binomial convolution There is also another binomial convolution in the mathematical literature
Counting Inversions. Proposition 1. - MIT Mathematics Counting Inversions Let Sn denote the set consisting of all permutations of [n] The inversion table of a permutation w 2 Sn is an n-tuple I(w) := (a1; : : : ; an), where ai denotes the number of elements j in w to the left of i with j > i Observe that
Binomial Transform -- from Wolfram MathWorld The inverse transform is a_n=sum_ (k=0)^n (n; k)b_k (Sloane and Plouffe 1995, pp 13 and 22) The inverse binomial transform of b_n=1 for prime n and b_n=0 for composite n is 0, 1, 3, 6, 11, 20, 37, 70,
Sun’s Binomial Inversion Formula - IDOSI Sun’s Binomial Inversion Formula 1A Lucas-Bravo, 2J López-Bonilla and 2S Vidal-Beltrán 1UPIITA, Instituto Politécnico Nacional, Av IPN 2580, Col Barrio la Laguna 07340, CDMX, México 2ESIME-Zacatenco, Instituto Politécnico Nacional, Edif 4, 1er Piso, Col Lindavista CP 07738 CDMX, México