Monomials with degree k of the following polynomial ( 1 + <mrow class="MJX-TeXAtom-ORD

Sovardipk

Sovardipk

Answered question

2022-06-29

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

Answer & Explanation

Dayana Zuniga

Dayana Zuniga

Beginner2022-06-30Added 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
= 0 n s ( 1 ) ( n s ) ( k s d 1 n s 1 ) .
Plugging all together, we get
s = 0 n s = 0 k r = 0 s = 0 n s ( 1 ) r + ( n s ) ( s r ) ( s ( q + 1 ) r + s 1 s 1 ) ( n s ) ( k s d 1 n s 1 )
Not entirely sure if this sum simplifies.

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