cp's OEIS Frontend

A329861 Triangle read by rows where T(n,k) is the number of compositions of n with cuts-resistance k.

%I A329861 #6 Nov 24 2019 09:59:45
%S A329861 1,0,1,0,1,1,0,3,0,1,0,4,3,0,1,0,7,6,2,0,1,0,14,9,6,2,0,1,0,23,22,10,
%T A329861 6,2,0,1,0,39,47,22,10,7,2,0,1,0,71,88,52,24,10,8,2,0,1,0,124,179,101,
%U A329861 59,26,11,9,2,0,1,0,214,354,220,112,71,28,12,10,2,0,1
%N A329861 Triangle read by rows where T(n,k) is the number of compositions of n with cuts-resistance k.
%C A329861 A composition of n is a finite sequence of positive integers summing to n.
%C A329861 For the operation of shortening all runs by 1, cuts-resistance is defined as the number of applications required to reach an empty word.
%e A329861 Triangle begins:
%e A329861   1
%e A329861   0  1
%e A329861   0  1  1
%e A329861   0  3  0  1
%e A329861   0  4  3  0  1
%e A329861   0  7  6  2  0  1
%e A329861   0 14  9  6  2  0  1
%e A329861   0 23 22 10  6  2  0  1
%e A329861   0 39 47 22 10  7  2  0  1
%e A329861   0 71 88 52 24 10  8  2  0  1
%e A329861 Row n = 6 counts the following compositions (empty columns not shown):
%e A329861   (6)     (33)    (222)    (11112)  (111111)
%e A329861   (15)    (114)   (1113)   (21111)
%e A329861   (24)    (411)   (3111)
%e A329861   (42)    (1122)  (11121)
%e A329861   (51)    (1131)  (11211)
%e A329861   (123)   (1221)  (12111)
%e A329861   (132)   (1311)
%e A329861   (141)   (2112)
%e A329861   (213)   (2211)
%e A329861   (231)
%e A329861   (312)
%e A329861   (321)
%e A329861   (1212)
%e A329861   (2121)
%t A329861 degdep[q_]:=Length[NestWhileList[Join@@Rest/@Split[#]&,q,Length[#]>0&]]-1;
%t A329861 Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],degdep[#]==k&]],{n,0,10},{k,0,n}]
%Y A329861 Row sums are A000079.
%Y A329861 Column k = 1 is A003242 (for n > 0).
%Y A329861 Column k = 2 is A329863.
%Y A329861 Row sums without the k = 1 column are A261983.
%Y A329861 The version for runs-resistance is A329744.
%Y A329861 The version for binary vectors is A329860.
%Y A329861 The cuts-resistance of the binary expansion of n is A319416.
%Y A329861 Cf. A175342, A240085, A319411, A319420, A319421, A329738, A329862, A329865.
%K A329861 nonn,tabl
%O A329861 0,8
%A A329861 _Gus Wiseman_, Nov 23 2019