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 A106356 Triangle T(n,k) 0<=k
 1, 1, 1, 3, 0, 1, 4, 3, 0, 1, 7, 6, 2, 0, 1, 14, 7, 8, 2, 0, 1, 23, 20, 10, 8, 2, 0, 1, 39, 42, 22, 13, 9, 2, 0, 1, 71, 72, 58, 28, 14, 10, 2, 0, 1, 124, 141, 112, 72, 33, 16, 11, 2, 0, 1, 214, 280, 219, 150, 92, 36, 18, 12, 2, 0, 1, 378, 516, 466, 311, 189, 112, 40, 20, 13, 2, 0, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS For n > 0, also the number of compositions of n with k + 1 maximal anti-runs (sequences without adjacent equal terms). - Gus Wiseman, Mar 23 2020 LINKS Alois P. Heinz, Rows n = 1..141, flattened A. Knopfmacher and H. Prodinger, On Carlitz compositions, European Journal of Combinatorics, Vol. 19, 1998, pp. 579-589. EXAMPLE T(4,1) = 3 because the compositions of 4 with 1 adjacent equal part are 1+1+2, 2+1+1, 2+2. Triangle begins: 1; 1,   1; 3,   0,  1; 4,   3,  0, 1; 7,   6,  2, 0, 1; 14,  7,  8, 2, 0, 1; 23, 20, 10, 8, 2, 0, 1; From Gus Wiseman, Mar 23 2020 (Start) Row n = 6 counts the following compositions (empty column shown by dot):   (6)     (33)    (222)    (11112)  .  (111111)   (15)    (114)   (1113)   (21111)   (24)    (411)   (1122)   (42)    (1131)  (2211)   (51)    (1221)  (3111)   (123)   (1311)  (11121)   (132)   (2112)  (11211)   (141)           (12111)   (213)   (231)   (312)   (321)   (1212)   (2121) (End) MAPLE b:= proc(n, h, t) option remember;       if n=0 then `if`(t=0, 1, 0)     elif t<0 then 0     else add(b(n-j, j, `if`(j=h, t-1, t)), j=1..n)       fi     end: T:= (n, k)-> b(n, -1, k): seq(seq(T(n, k), k=0..n-1), n=1..15); # Alois P. Heinz, Oct 23 2011 MATHEMATICA b[n_, h_, t_] := b[n, h, t] = If[n == 0, If[t == 0, 1, 0], If[t<0, 0, Sum[b[n-j, j, If [j == h, t-1, t]], {j, 1, n}]]]; T[n_, k_] := b[n, -1, k]; Table[Table[T[n, k], {k, 0, n-1}], {n, 1, 15}] // Flatten (* Jean-François Alcover, Feb 20 2015, after Alois P. Heinz *) Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], n==0||Length[Split[#, #1!=#2&]]==k+1&]], {n, 0, 12}, {k, 0, n}] (* Gus Wiseman, Mar 23 2020 *) CROSSREFS Row sums: 2^(n-1)=A000079(n-1). Columns 0-4: A003242, A106357-A106360. The version counting adjacent unequal parts is A238279. The k-th composition in standard-order has A124762(k) adjacent equal parts and A333382(k) adjacent unequal parts. The k-th composition in standard-order has A124767(k) maximal runs and A333381(k) maximal anti-runs. The version for ascents/descents is A238343. The version for weak ascents/descents is A333213. Cf. A064113, A066099, A233564, A333214, A333216. Sequence in context: A256987 A048963 A119458 * A091613 A039727 A137176 Adjacent sequences:  A106353 A106354 A106355 * A106357 A106358 A106359 KEYWORD nonn,tabl AUTHOR Christian G. Bower, Apr 29 2005 STATUS approved

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Last modified November 27 18:22 EST 2020. Contains 338683 sequences. (Running on oeis4.)