Sovardipk

Sovardipk

Answered

2022-06-29

Monomials with degree k of the following polynomial ( 1 + 𝑥 1 + ⋯ + 𝑥 1 q + y 1 + ⋯ + y 1 d ) n

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Answer & Explanation

Dayana Zuniga

Dayana Zuniga

Expert

2022-06-30Added 16 answers

Move s and s′ as you wish as
∑ s = 0 n ( n s ) ∑ s ′ = 0 k ( # ways  x i  add to s' ) ⋅ ( # ways  y j  add to k-s' ) ,
using stars and bars(and inclusion-exclusion), we know that the number of tuples adding to s′ with parts less or equal than q is
∑ r = 0 s ( − 1 ) r ( s r ) ( s ′ − ( q + 1 ) r + s − 1 s − 1 ) ,
similarly for the y j notice that we are not allowing 0 as a part (I took 1 = x i 0 ) and we have
∑ ℓ = 0 n − s ( − 1 ) ℓ ( n − s ℓ ) ( k − s ′ − d ℓ − 1 n − s − 1 ) .
Plugging all together, we get
∑ s = 0 n ∑ s ′ = 0 k ∑ r = 0 s ∑ ℓ = 0 n − s ( − 1 ) r + ℓ ( n s ) ( s r ) ( s ′ − ( q + 1 ) r + s − 1 s − 1 ) ( n − s ℓ ) ( k − s ′ − d ℓ − 1 n − s − 1 )
Not entirely sure if this sum simplifies.

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