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%I #5 Nov 21 2019 22:15:22
%S 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,
%T 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,
%U 14,202,367,226,213,1,1,0,0,0,0,0,52,469,750,376,395,5,1
%N Triangle read by rows where T(n,k) is the number of compositions of n >= 1 with runs-resistance n - k, 1 <= k <= n.
%C A composition of n is a finite sequence of positive integers with sum n.
%C 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.
%e Triangle begins:
%e 1
%e 1 1
%e 2 1 1
%e 2 3 2 1
%e 2 6 6 1 1
%e 0 4 9 15 3 1
%e 0 2 16 22 22 1 1
%e 0 0 8 37 38 41 3 1
%e 0 0 0 26 86 69 72 2 1
%e 0 0 0 2 78 175 124 129 3 1
%e 0 0 0 0 14 202 367 226 213 1 1
%e 0 0 0 0 0 52 469 750 376 395 5 1
%e Row n = 6 counts the following compositions:
%e (1,1,3,1) (1,1,4) (1,5) (3,3) (6)
%e (1,3,1,1) (4,1,1) (2,4) (2,2,2)
%e (1,1,1,2,1) (1,1,1,3) (4,2) (1,1,1,1,1,1)
%e (1,2,1,1,1) (1,2,2,1) (5,1)
%e (2,1,1,2) (1,2,3)
%e (3,1,1,1) (1,3,2)
%e (1,1,1,1,2) (1,4,1)
%e (1,1,2,1,1) (2,1,3)
%e (2,1,1,1,1) (2,3,1)
%e (3,1,2)
%e (3,2,1)
%e (1,1,2,2)
%e (1,2,1,2)
%e (2,1,2,1)
%e (2,2,1,1)
%t runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
%t Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],runsres[#]==n-k&]],{n,10},{k,n}]
%Y Row sums are A000079.
%Y Column sums are A329768.
%Y The version with rows reversed is A329744.
%Y Cf. A000740, A008965, A098504, A242882, A318928, A329745, A329746, A329747, A329767.
%K nonn,tabl
%O 1,4
%A _Gus Wiseman_, Nov 21 2019