05AB1E, 10 9 bytes

œÙʒη€àÙgQ

Try it online or verify (almost) all test cases (test case [1,2,3,4,5,6,7,8,9],4 times out).

Footer of the TIO does what OP asked at the bottom:

Optional: If possible, can you add something like if length is greater than 20: print length else print answer. In the footer, not in the code.

Explanation:

œ            # Permutations of the (implicit) input-list

             #  i.e. [1,2,2] → [[1,2,2],[1,2,2],[2,1,2],[2,2,1],[2,1,2],[2,2,1]]

 Ù           # Only leave the unique permutations

             #  i.e. [[1,2,2],[1,2,2],[2,1,2],[2,2,1],[2,1,2],[2,2,1]]

             #   → [[1,2,2],[2,1,2],[2,2,1]]

  ʒ          # Filter it by:

   η         #  Push the prefixes of the current permutation

             #   i.e. [1,2,2] → [[1],[1,2],[1,2,2]]

    ۈ       #  Calculate the maximum of each permutation

             #   i.e. [[1],[1,2],[1,2,2]] → [1,2,2]

      Ù      #  Only leave the unique maximums

             #   i.e. [1,2,2] → [1,2]

       g     #  And take the length

             #   i.e. [1,2] → 2

        Q    #  And only leave those equal to the second (implicit) input

             #   i.e. 2 and 2 → 1 (truthy)

Haskell, 73 bytes

import Data.List

f n=filter((n==).length.nub.scanl1 max).nub.permutations

Try it online!

Jelly, 12 10 bytes

Œ!Q»QL=ʋƇ

Try it online!

-2 bytes by @Erik the Outgolfer

This is a dyadic function taking the building heights and k in that order.

Œ!                All permutations of first input

Œ!Q               Unique permutations of first input

   »              Running maximum

     Q             Unique values

      L            Length of this array

       =           Equals k

        ʋ        Create a monad from these 4 links

   »QL=ʋ        "Are exactly k buildings visible in arrangement x?"

         Ƈ     Filter if f(x)

Œ!Q»QL=ʋƇ     All distinct perms of first input with k visible buildings.

Pyth, 18 16 bytes

fqvzl{meSd._T{.p

Try it here.

Note that the online version of the Pyth interpreter throws a memory error on the largest test case.

f                       Filter lambda T:

  q                       Are second input and # visible buildings equal?

    v z                     The second input value

    l {                     The number of unique elements in

        m                   the maximums

          e S d             ...

          ._ T              of prefixes of T

    { .p                  over unique permutations of (implicit first input)

Perl 6, 81 63 bytes

-18 bytes thanks to nwellnhof!

{;*.permutations.unique(:with(*eqv*)).grep:{$_==set [max] @_}}

Try it online!

Anonymous code block that takes input curried, e.g. f(n)(list). That .unique(:with(*eqv*)) is annoyingly long though :(

Explanation:

{;                                                            }  # Anonymous code block

  *.permutations.unique(:with(*eqv*))  # From all distinct permutations

                                     .grep:{                 }  # Filter where

                                                set [max] @_   # Visible buildings

                                            $_==      # Equals num

Japt, 11 bytes

á f_åÔâ Ê¥V

Try it online!

For the longer outputs, adding } l to the end will output the length instead. The online interpreter times out for the [1,2,3,4,5,6,7,8,9],4 test case, regardless of outputting the length or the list.

Explanation:

á              :Get all permutations

  f_           :Keep only ones where:

    åÔ         : Get the cumulative maximums (i.e. the visible buildings)

      â Ê      : Count the number of unique items

         ¥V    : True if it's the requested number

JavaScript (ES6), 108 107 bytes

Takes input as (k)(array). Prints the results with alert().

k=>P=(a,p=,n=k,h=0)=>a.map((v,i)=>P(a.filter(_=>i--),[...p,v],n-(v>h),v>h?v:h))+a||n||P[p]||alert(P[p]=p)

Try it online!

Commented

k =>                        // k = target number of visible buildings

  P = (                     // P = recursive function taking:

    a,                      //   a = list of building heights

    p = ,                 //   p = current permutation

    n = k,                  //   n = counter initialized to k

    h = 0                   //   h = height of the highest building so far

  ) =>                      //

    a.map((v, i) =>         // for each value v at position i in a:

      P(                    //   do a recursive call:

        a.filter(_ => i--), //     using a copy of a without the i-th element

        [...p, v],          //     append v to p

        n - (v > h),        //     decrement n if v is greater than h

        v > h ? v : h       //     update h to max(h, v)

      )                     //   end of recursive call

    )                       // end of map()

    + a ||                  // unless a was not empty,

    n ||                    // or n is not equal to 0,

    P[p] ||                 // or p was already printed,

    alert(P[p] = p)         // print p and store it in P

Python 2, 114 113 bytes

lambda a,n:{p for p in permutations(a)if-~sum(p[i]>max(p[:i])for i in range(1,len(p)))==n}

from itertools import*

Try it online!

_{-1 byte, thanks to ovs}

Python 3, 113 bytes

lambda a,n:{p for p in permutations(a)if sum(v>max(p[:p.index(v)]+(v-1,))for v in{*p})==n}

from itertools import*

Try it online!

J, 43 38 bytes

-5 bytes after incorporating an optimization from Kevin's O5AB13 answer

(]#~[=([:#@~.>./)"1@])[:~.i.@!@#@]A.]

Try it online!

ungolfed

(] #~ [ = ([: #@~. >./)"1@]) ([: ~. i.@!@#@] A. ])

explanation

we're merely listing all possible perms i.@!@#@] A. ], taking uniq items thereof with ~., then filtering those by the number of visible building, which must equal the left input.

the key logic is in the parenthetical verb which calcs the number of visible buildings:

([: #@~. >./)

Here we use a max scan >./ to keep a tally of the tallest building seen so far. Then we just take the unique elements of the max scan, and that's the number of visible buildings.