Bijection between partitions Give a bijective mapping from the set of partitions of $[n]$ with no cyclically consecutive integers in a block, and the set of partitions  of $[n]$ with no singleton blocks.

All the mappings that I come up with are injective. Can somebody please help?
[1,2,3]       [1] [2] [3]
[1] [2,3]     [2] [1,3]
[2] [1,3]     [3] [1,2]
[3] [1,2]     [1] [2,3]
[1] [2] [3]                      [1,2,3]

[1,2,3,4]     [1] [2] [3] [4]
[1] [2,3,4]   [2] [3] [1,4]
[2] [1,3,4]   [3] [4] [1,2]
[3] [1,2,4]   [1] [4] [2,3]
[4] [1,2,3]   [1] [2] [3,4]
[1,2] [3,4]   [1] [3] [2,4]
[1,3] [2,4]                     [1,3] [2,4]
[1,4] [2,3]   [2] [4] [1,3]
[1] [2] [3,4]  [3] [1,2,4]
[1] [3] [2,4]                   [1,2] [3,4]
[1] [4] [2,3]  [2] [1,3,4]
[2] [3] [1,4]   [4] [1,2,3]
[2] [4] [1,3]                   [2,3] [4,1]
[3] [4] [1,2]   [1] [2,3,4]
[1] [2] [3] [4]                 [1 2 3 4]


Above is a grouping for the $n=3$ and $n=4$ case. The way I was approaching it was to think of a way of flagging cyclic consecutive integers in a group with singletons, and the rule I came up with for this purpose was to say that if $[i,i+1]$ appear in any given block, then in the bijective map $[i]$ would be by itself, and $[i+1]$ would be set aside and combined with the others (if $[i+1,i+2]$ is not in the same block). I believe this is a bijection between bad sets to bad sets, but I suppose there is something to be done for the other way :).

Looks like it's not clear how to design the map on good sets... The current pairing for $n=4$ above works as follows: If there are no singletons, the map is an identity, otherwise for each singleton $[i]$, put $i+1$ in the same group. If $[i+1]$ is also a singleton, grab $[i+2]$ and place in the same group, etc. As this is homework, I don't want to spoil all the beans, only to give some ideas to push (it could be this doesn't work out, but it's an idea).
I'll try to give a bijection in the $n=2, 3, 4$ cases to see if we can see something.

{1 2}        {1,2}

{1 2 3}      {1,2,3}

{1 2 3 4}    {1,2,3,4}
{1 3,2 4}    {1 3,2 4}
{2 1,3 4}    {2,1 3,4}
{1 4,2 3}    {1,4 2,3}


These bijections look "natural", but I can't really think of a way to expand the thinking...