Then we use the expression for the power sum polynomial as an alternating sum of hook Schur polynomials.
We next rewrite the power sum symmetric function as an alternating sum of hook Schur functions
We push forward the resulting alternating sum to [[LAMBDA].sub.k,n], and then consider the class in q[H.sup.*](Gr(k, n)).
The main ingredient is Rice's formula Flajolet and Sedgewick (1995) which allows to write an alternating sum as a contour integral:
For the asymptotic evaluation, we use a contour integral representation of alternating sums ("Rice's method").
Proof of Theorem 2: The exactness of the sequence (1) gives us that the alternating sum
of dimensions is 0.
The terms [alpha] and [beta] are typical expressions to which 'Rice's method' can be applied, due to the presence of the binomial coefficient inside an alternating sum. First we deal with a, and the function involved for Rice's method is f(z) = 1/[Q.sup.z-1]-1.
Now we look at the alternating sum labelled (b) which has a double pole at z = 1 and a simple pole at z = 2.
Now Rice's method can again be used to approximate the alternating sums. First we deal with part (a) by considering the poles of the function [n;z] f(z) with f(z) = [Q.sup.z]-1/1-[Q.sup.z-2].
Turker, "Alternating sums
of the powers of Fibonacci and Lucas numbers," Miskolc Mathematical Notes, vol.
Simsek, "A note on the alternating sums
of powers of consecutive q-integers," Advanced Studies in Contemporary Mathematics, vol.
They cover initial encounters with combinatorial reasoning; selections, arrangements, and distributions; binomial series and generating functions; alternating sums
, the inclusion-exclusion principle, rook polynomials, and Fibonacci Nim; recurrence relations; special numbers; linear spaces and recurrence sequences; and counting with symmetries.