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A329767 Triangle read by rows where T(n,k) is the number of binary words of length n >= 0 with runs-resistance k, 0 <= k <= n.

%I A329767 #9 Nov 30 2019 09:10:15
%S A329767 1,2,0,0,2,2,0,2,2,4,0,2,4,6,4,0,2,2,12,12,4,0,2,6,30,18,8,0,0,2,2,44,
%T A329767 44,32,4,0,0,2,6,82,76,74,16,0,0,0,2,4,144,138,172,52,0,0,0,0,2,6,258,
%U A329767 248,350,156,4,0,0,0,0,2,2,426,452,734,404,28,0,0,0,0
%N A329767 Triangle read by rows where T(n,k) is the number of binary words of length n >= 0 with runs-resistance k, 0 <= k <= n.
%C A329767 A composition of n is a finite sequence of positive integers with sum n.
%C A329767 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.
%C A329767 Except for the k = 0 column and the n = 0 and n = 1 rows, this is the triangle appearing on page 3 of Lenormand, which is A319411. Unlike A318928, we do not here require that a(n) >= 1.
%C A329767 The n = 0 row is chosen to ensure that the row-sums are A000079, although the empty word arguably has indeterminate runs-resistance.
%H A329767 Claude Lenormand, <a href="/A318921/a318921.pdf">Deux transformations sur les mots</a>, Preprint, 5 pages, Nov 17 2003.
%e A329767 Triangle begins:
%e A329767    1
%e A329767    2   0
%e A329767    0   2   2
%e A329767    0   2   2   4
%e A329767    0   2   4   6   4
%e A329767    0   2   2  12  12   4
%e A329767    0   2   6  30  18   8   0
%e A329767    0   2   2  44  44  32   4   0
%e A329767    0   2   6  82  76  74  16   0   0
%e A329767    0   2   4 144 138 172  52   0   0   0
%e A329767    0   2   6 258 248 350 156   4   0   0   0
%e A329767    0   2   2 426 452 734 404  28   0   0   0   0
%e A329767 For example, row n = 4 counts the following words:
%e A329767   0000  0011  0001  0010
%e A329767   1111  0101  0110  0100
%e A329767         1010  0111  1011
%e A329767         1100  1000  1101
%e A329767               1001
%e A329767               1110
%t A329767 runsres[q_]:=If[Length[q]==1,0,Length[NestWhileList[Length/@Split[#]&,q,Length[#]>1&]]-1];
%t A329767 Table[Length[Select[Tuples[{0,1},n],runsres[#]==k&]],{n,0,10},{k,0,n}]
%Y A329767 Row sums are A000079.
%Y A329767 Column k = 2 is A319410.
%Y A329767 Column k = 3 is 2 * A329745.
%Y A329767 The version for compositions is A329744.
%Y A329767 The version for partitions is A329746.
%Y A329767 The number of nonzero entries in row n > 0 is A319412(n).
%Y A329767 The runs-resistance of the binary expansion of n is A318928.
%Y A329767 Cf. A001037, A096365, A225485, A245563, A319411, A325280, A329747, A329750.
%K A329767 nonn,tabl
%O A329767 0,2
%A A329767 _Gus Wiseman_, Nov 21 2019