### A Combinatorial Problem from HMMT-2009Given a rearrangement of the numbers from 1 to n, each pair of consecutive elements a and b of the sequence can be either increasing or decreasing. How many rearrangements have of the numbers from 1 to n have exactly two increasing pairs?

Meera Sunker 2022-05-20

### 1a.  Show from first principles, i.e., by using the definition of linear independence,that if μ = x + iy, y ̸= 0 is an eigenvalue of a real matrixA with associated eigenvector v = u + iw, then the two real solutionsY(t) = eat(u cos bt − wsin bt)andZ(t) = eat(u sin bt + wcos bt)are linearly independent solutions of ˙X = AX. 1b. Use (a) to solve the system (see image)

Meera Sunker 2022-05-20