2 questions of Combinatorics 1) Let m, n E Z with 0 < m < 7/

Math Oct 06, 2021

2 questions of Combinatorics

1) Let m, n E Z with 0 < m < 7/. Give a combinatorial proof of the equality
nx—.—nt in n n 9 /\ m i 1-t — i) = 7) 1—m1 • i=0
No algebraic manipulations are allowed; you must give combinatorial interpretations of both sides in the way that they are written.

Q4 (10 points)
Let n E N. Use lattice paths to give a combinatorial proof of the identity
(2n \ k n) 1 n-1 1 (2n - 2k - 1\ n - k ) . (21C (2n) n+1 n 4-.0 k 1 k k ) k=0 +