Pure function inside another pure function

This is probably not going to be the best answer but offering it as an opener or as a guide to towards a better solution

Setting your initial input as a function

f[n_]:=Length[Select[IntegerPartitions[10],First[#]==n&]]

then

Map[f,Range[10]]

{1, 5, 8, 9, 7, 5, 3, 2, 1, 1}

No doubt regular contributors can improve on this

You can have a pure function inside a pure function even in this case, you just can't have the name of the parameter being "#" in both. This works:

Map[Function[x, 
  Length[Select[IntegerPartitions[10], First[#] == x &]]], Range[10]]

Using Curry or OperatorApplied

Pure function nesting is one of the use cases for Curry, introduced in version 11.3:

Length[
  Select[IntegerPartitions[10], Curry[First[#] == #2 &][#]]
] & /@ Range[10]

(* {1, 5, 8, 9, 7, 5, 3, 2, 1, 1} *)

First[#1] == #2 & defines a two argument pure function. By wrapping Curry[...][#] around this, we bind the inner slot reference #2 to the value of the outer slot reference #. After binding, we are left with a single argument function that is suitable for use with Select. For example, in one mapping iteration the outer # has the value 7 and the curried function Curry[First[#1] == #2 &][7] is essentially equivalent to First[#1] == 7 &.

Update for version 12.1

In version 12.1, OperatorApplied is the new name for Curry. The documentation for Curry states that it is now being phased out.

Currying The Hard Way

A rather obscure idiom makes it possible to perform such currying in pure functions, even before version 11.3:

Length[
  Select[IntegerPartitions[10], #0[[0]][First[#[1]] == #2] &[Slot, #]]
] & /@ Range[10]

(* {1, 5, 8, 9, 7, 5, 3, 2, 1, 1} *)

The cryptic expression #0[[0]][First[#[1]] == #2]&[Slot, #] performs the necessary currying. #0[[0]] expands to Function. The disguised use of the symbol Function is necessary to avoid shadowing the inner slot references. #[1] expands to Slot[1], i.e. #1. #2 expands to the value of the outer #. So, if the outer # were 7, then we would get First[#1] == 7 & as desired.

This construction is likely too ugly to actually use. Thank goodness for the arrival of Curry :)