A329750 Triangle read by rows where T(n,k) is the number of compositions of n >= 1 with runs-resistance n - k, 1 <= k <= n.

1, 1, 1, 2, 1, 1, 2, 3, 2, 1, 2, 6, 6, 1, 1, 0, 4, 9, 15, 3, 1, 0, 2, 16, 22, 22, 1, 1, 0, 0, 8, 37, 38, 41, 3, 1, 0, 0, 0, 26, 86, 69, 72, 2, 1, 0, 0, 0, 2, 78, 175, 124, 129, 3, 1, 0, 0, 0, 0, 14, 202, 367, 226, 213, 1, 1, 0, 0, 0, 0, 0, 52, 469, 750, 376, 395, 5, 1

Offset: 1

Author

Gus Wiseman, Nov 21 2019

Keywords

• nonn
• tabl

Comments

A composition of n is a finite sequence of positive integers with sum n.

For the operation of taking the sequence of run-lengths of a finite sequence, runs-resistance is defined as the number of applications required to reach a singleton.

Examples

			Triangle begins:
   1
   1   1
   2   1   1
   2   3   2   1
   2   6   6   1   1
   0   4   9  15   3   1
   0   2  16  22  22   1   1
   0   0   8  37  38  41   3   1
   0   0   0  26  86  69  72   2   1
   0   0   0   2  78 175 124 129   3   1
   0   0   0   0  14 202 367 226 213   1   1
   0   0   0   0   0  52 469 750 376 395   5   1
Row n = 6 counts the following compositions:
  (1,1,3,1)    (1,1,4)      (1,5)      (3,3)          (6)
  (1,3,1,1)    (4,1,1)      (2,4)      (2,2,2)
  (1,1,1,2,1)  (1,1,1,3)    (4,2)      (1,1,1,1,1,1)
  (1,2,1,1,1)  (1,2,2,1)    (5,1)
               (2,1,1,2)    (1,2,3)
               (3,1,1,1)    (1,3,2)
               (1,1,1,1,2)  (1,4,1)
               (1,1,2,1,1)  (2,1,3)
               (2,1,1,1,1)  (2,3,1)
                            (3,1,2)
                            (3,2,1)
                            (1,1,2,2)
                            (1,2,1,2)
                            (2,1,2,1)
                            (2,2,1,1)
		

Crossrefs

Row sums are A000079.

Column sums are A329768.

The version with rows reversed is A329744.

Cf. A000740, A008965, A098504, A242882, A318928, A329745, A329746, A329747, A329767.

Programs

Mathematica

runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],runsres[#]==n-k&]],{n,10},{k,n}]