Below  T(n, k)  denotes the number of permutations on  n  symbols 
decomposing consecutively into just  k  separate permutations, see 
A059438, A263484;  a(n) = T(n, 1)  denotes number indecomposable, 
see A003319. 

  Kindly colleagues have pointed out to me an efficient algorithm 
for these numbers, under OEIS entry A059438 (sans row-reversal).  
Since the Mathematica program supplied there is not straightforward 
to decipher nor to motivate, a short discussion seems appropriate. 

  Once pointed out it becomes obvious that every permutation, if  
decomposing into  k  parts, also decomposes into two permutations 
with  k-l  and  l  parts respectively, for any  0 <= l <= k . 
In particular, setting  l = 1  yields the convolution: for  k > 1 , 

    T(n,k)  =  \sum_{k-1 <= i <= n-1} T(i,k-1) a(n-i) ; 

where  a(n) = T(n, 1)  may be computed separately for  n > 0  via: 

    a(n)  =  n! - \sum_{1 <= i <= n-1} i! a(n-i) . 

  So in a nutshell  T(n,k)  equals the  n-shifted  k-fold convolution 
of  a(n) .  But that is a little tricky to spot immediately, because  
T(n,n-k)  happens to be polynomial in  n  of degree  k : a natural 
though fruitless initial response is to focus on these deceptively 
promising polynomials instead, suggesting a presumptive reason for 
the existence of mysteriously row-reversed A263484 in the first place. 

  The reliance on  a(n)  may furthermore be eliminated by noticing 
that row  n  sums to  n! ; whence  for  n > 0 , 

    T(n,1)  =  n! - \sum_{2 <= i <= n} T(n,j) . 

  Attention to argument support and evaluation order now produces 
the Maple program below, reasonably readable --- albeit compromised 
by logically vacuous "proc() ... end()" brackets necessary to 
circumvent one of Maple's numerous syntactic idiosyncrasies! 
Clean copy available at 
    https://www.dropbox.com/s/8tdz9499suazcd2/A263484_%26_A059438.txt

Fred Lunnon [16/06/19] 

############################################################

restart; nmax := 12; 

krodel := proc(p, q) if p = q then 1 else 0 fi end; 

for n from 1 to nmax do 
  for k from n by -1 to 1 do 
    T[n][k] := proc() 
      if k = 0 and n >= k then krodel(n, k) 
      elif k = 1 then n! - add(T[n][k], k = 2..n) 
      else add( T[i][k-1]*T[n-i][1], i = k-1..n-1) fi 
    end() od od: 

[seq( [seq( T[n][k], k = 1..n)], n = 1..nmax)];  # triangle 
[seq( seq( T[n][k], k = 1..n), n = 1..nmax)];  # A059438 
[seq( seq( T[n][n+1-k], k = 1..n), n = 1..nmax)];  # A263484 
[seq( T[n][1], n = 1..nmax)];  # A003319  

############################################################

[
  [1], 
  [1, 1], 
  [3, 2, 1], 
  [13, 7, 3, 1], 
  [71, 32, 12, 4, 1], 
  [461, 177, 58, 18, 5, 1], 
  [3447, 1142, 327, 92, 25, 6, 1], 
  [29093, 8411, 2109, 531, 135, 33, 7, 1], 
  [273343, 69692, 15366, 3440, 800, 188, 42, 8, 1], 
  [2829325, 642581, 125316, 24892, 5226, 1146, 252, 52, 9, 1], 
  [31998903, 6534978, 1135329, 200344, 37690, 7572, 1582, 328, 63, 10, 1], 
  [392743957, 72754927, 11348623, 1786101, 300170, 54598, 10598, 2122, 417, 75, 11, 1], 
NULL];