Confused with HoldForm in Postfix notation

Question

Why does the following code produce different result? In my mental model, they should be the same.

Table[With[{x = i^k}, HoldForm[x]], {k, 1, 5}]
With[{x = i^k}, HoldForm[x]] // Table[#, {k, 1, 5}] &

Output:

{i,i^2,i^3,i^4,i^5}
{i^k,i^k,i^k,i^k,i^k}

Answer 1

The difference is because in the second case

With[{x = i^k}, HoldForm[x]] // Table[#, {k, 1, 5}] &

The With expression is evaluated (giving HoldForm[i^k]) before it is substituted inside the placeholder in Table. To prevent thise pre-evaluation wrap it around Unevaluated as follows:

Unevaluated[With[{x = i^k}, HoldForm[x]]] // Table[#, {k, 1, 5}] &

Which gives as before:

{i, i^2, i^3, i^4, i^5}

Answer 2

To be more specific, the difference is not because of the postfix application but because of the pure function (the part with # and &) application:

Table[With[{x = i^k}, HoldForm[x]], {k, 1, 5}] (* no pure function *)
(* ==> {i, i^2, i^3, i^4, i^5} *)

With[{x = i^k}, HoldForm[x]] // (Table[#, {k, 1, 5}] &)  (* postfix pure function *)
(* ==> {i^k, i^k, i^k, i^k, i^k} *)

(Table[#, {k, 1, 5}] &) @ With[{x = i^k}, HoldForm[x]]  (* prefix pure function *)
(* ==> {i^k, i^k, i^k, i^k, i^k} *)

Accordingly, if you simplify your example, and put some Prints in it:

Block[{k = 1}, Print[2]; With[{x = k}, Print[1]; HoldForm@x]]

During evaluation of In[17]:= 2
During evaluation of In[17]:= 1
(* ==> 1 *)

Block[{k = 1}, Print[2]; #] &@With[{x = k}, Print[1]; HoldForm@x]

During evaluation of In[17]:= 1
During evaluation of In[17]:= 2
(* ==> k *)

As you can see, in the last case, the With is evaluated first, not Block, resulting in thus a replacement x -> k, so Block cannot replace x in the second step as there is no x anymore in the expression.

An even more simple example that shows the reversed evaluation sequence for pure function application compared to normal, nested expression evaluation:

(Print[2]; Print[1];) 

2
1

(Print[2]; #;) & @ Print[1]; (* or:  Print[1]; // (Print[2]; #;) & *)

1
2