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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.

%I A329750 #5 Nov 21 2019 22:15:22
%S A329750 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 A329750 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 A329750 14,202,367,226,213,1,1,0,0,0,0,0,52,469,750,376,395,5,1
%N 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.
%C A329750 A composition of n is a finite sequence of positive integers with sum n.
%C A329750 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 A329750 Triangle begins:
%e A329750    1
%e A329750    1   1
%e A329750    2   1   1
%e A329750    2   3   2   1
%e A329750    2   6   6   1   1
%e A329750    0   4   9  15   3   1
%e A329750    0   2  16  22  22   1   1
%e A329750    0   0   8  37  38  41   3   1
%e A329750    0   0   0  26  86  69  72   2   1
%e A329750    0   0   0   2  78 175 124 129   3   1
%e A329750    0   0   0   0  14 202 367 226 213   1   1
%e A329750    0   0   0   0   0  52 469 750 376 395   5   1
%e A329750 Row n = 6 counts the following compositions:
%e A329750   (1,1,3,1)    (1,1,4)      (1,5)      (3,3)          (6)
%e A329750   (1,3,1,1)    (4,1,1)      (2,4)      (2,2,2)
%e A329750   (1,1,1,2,1)  (1,1,1,3)    (4,2)      (1,1,1,1,1,1)
%e A329750   (1,2,1,1,1)  (1,2,2,1)    (5,1)
%e A329750                (2,1,1,2)    (1,2,3)
%e A329750                (3,1,1,1)    (1,3,2)
%e A329750                (1,1,1,1,2)  (1,4,1)
%e A329750                (1,1,2,1,1)  (2,1,3)
%e A329750                (2,1,1,1,1)  (2,3,1)
%e A329750                             (3,1,2)
%e A329750                             (3,2,1)
%e A329750                             (1,1,2,2)
%e A329750                             (1,2,1,2)
%e A329750                             (2,1,2,1)
%e A329750                             (2,2,1,1)
%t A329750 runsres[q_]:=Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1;
%t A329750 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],runsres[#]==n-k&]],{n,10},{k,n}]
%Y A329750 Row sums are A000079.
%Y A329750 Column sums are A329768.
%Y A329750 The version with rows reversed is A329744.
%Y A329750 Cf. A000740, A008965, A098504, A242882, A318928, A329745, A329746, A329747, A329767.
%K A329750 nonn,tabl
%O A329750 1,4
%A A329750 _Gus Wiseman_, Nov 21 2019