Partitions
Show that the number of partitions of an integer n into summands
of even size...
Partitions
Show that the number of partitions of an integer n into summands
of even size is equal to the number of partitions into summands
such that each summand occurs an even number of times.
For all integers n > 2, show that the number of integer
partitions of n in which each part is greater than one is given by
p(n)-p(n-1), where p(n) is the number of integer partitions of
n.
(a) Let n = 2k be an even integer. Show that x = rk
is an element of order 2 which commutes with every element of
Dn.
(b) Let n = 2k be an even integer. Show that x = rk
is the unique non-identity element which commutes with every
element of Dn.
(c) If n is an odd integer, show that the identity is the only
element of Dn which commutes with every element of
Dn.
Let N(n) be the number of all partitions of [n] with no
singleton blocks. And let A(n) be the number of all partitions of
[n] with at least one singleton block. Prove that for all n ≥ 1,
N(n+1) = A(n). Hint: try to give (even an informal) bijective
argument.
4. Let n ≥ 8 be an even integer and let k be an integer with 2 ≤
k ≤ n/2. Consider k-element subsets of the set S = {1, 2, . . . ,
n}. How many such subsets contain at least two even numbers?
Consider an n×n square board, where n is a fixed even positive
integer. The board is divided into n 2 unit squares. We say that
two different squares on the board are adjacent if they have a
common side. N unit squares on the board are marked in such a way
that every unmarked square on the board is adjacent to at least one
marked square. Determine the smallest possible value of N.