Find the number of partitions of the set
![\{1, 2, \cdots, n\}](/media/m/a/6/3/a630f5adc46e3b55073581772012ccf6.png)
into three subsets
![A_1,A_2,A_3](/media/m/1/4/0/1405261f4e0368095c427c4c5e9d0316.png)
, some of which may be empty, such that the following conditions are satisfied:
![(i)](/media/m/2/b/9/2b900b6092ba26af1415020bbd427e84.png)
After the elements of every subset have been put in ascending order, every two consecutive elements of any subset have different parity.
![(ii)](/media/m/a/6/5/a65ca45870cb4c928be31b9a8a6fc8e5.png)
If
![A_1,A_2,A_3](/media/m/1/4/0/1405261f4e0368095c427c4c5e9d0316.png)
are all nonempty, then in exactly one of them the minimal number is even .
Proposed by Poland.
%V0
Find the number of partitions of the set $\{1, 2, \cdots, n\}$ into three subsets $A_1,A_2,A_3$, some of which may be empty, such that the following conditions are satisfied:
$(i)$ After the elements of every subset have been put in ascending order, every two consecutive elements of any subset have different parity.
$(ii)$ If $A_1,A_2,A_3$ are all nonempty, then in exactly one of them the minimal number is even .
Proposed by Poland.