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A106356 Triangle T(n,k) 0<=k : Number of compositions of n with k adjacent equal parts.

%I A106356 #24 May 08 2025 04:19:58
%S A106356 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,
%T A106356 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,
%U A106356 280,219,150,92,36,18,12,2,0,1,378,516,466,311,189,112,40,20,13,2,0,1
%N A106356 Triangle T(n,k) 0<=k<n : Number of compositions of n with k adjacent equal parts.
%C A106356 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
%H A106356 Alois P. Heinz, <a href="/A106356/b106356.txt">Rows n = 1..141, flattened</a>
%H A106356 A. Knopfmacher and H. Prodinger, <a href="http://dx.doi.org/10.1006/eujc.1998.0216">On Carlitz compositions</a>, European Journal of Combinatorics, Vol. 19, 1998, pp. 579-589.
%e A106356 T(4,1) = 3 because the compositions of 4 with 1 adjacent equal part are 1+1+2, 2+1+1, 2+2.
%e A106356 Triangle begins:
%e A106356    1;
%e A106356    1,  1;
%e A106356    3,  0,  1;
%e A106356    4,  3,  0, 1;
%e A106356    7,  6,  2, 0, 1;
%e A106356   14,  7,  8, 2, 0, 1;
%e A106356   23, 20, 10, 8, 2, 0, 1;
%e A106356   ...
%e A106356 From _Gus Wiseman_, Mar 23 2020 (Start)
%e A106356 Row n = 6 counts the following compositions (empty column shown by dot):
%e A106356   (6)     (33)    (222)    (11112)  .  (111111)
%e A106356   (15)    (114)   (1113)   (21111)
%e A106356   (24)    (411)   (1122)
%e A106356   (42)    (1131)  (2211)
%e A106356   (51)    (1221)  (3111)
%e A106356   (123)   (1311)  (11121)
%e A106356   (132)   (2112)  (11211)
%e A106356   (141)           (12111)
%e A106356   (213)
%e A106356   (231)
%e A106356   (312)
%e A106356   (321)
%e A106356   (1212)
%e A106356   (2121)
%e A106356 (End)
%p A106356 b:= proc(n, h, t) option remember;
%p A106356       if n=0 then `if`(t=0, 1, 0)
%p A106356     elif t<0 then 0
%p A106356     else add(b(n-j, j, `if`(j=h, t-1, t)), j=1..n)
%p A106356       fi
%p A106356     end:
%p A106356 T:= (n, k)-> b(n, -1, k):
%p A106356 seq(seq(T(n, k), k=0..n-1), n=1..15); # _Alois P. Heinz_, Oct 23 2011
%t A106356 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_ *)
%t A106356 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 *)
%Y A106356 Row sums: 2^(n-1)=A000079(n-1). Columns 0-4: A003242, A106357-A106360.
%Y A106356 The version counting adjacent unequal parts is A238279.
%Y A106356 The k-th composition in standard-order has A124762(k) adjacent equal parts and A333382(k) adjacent unequal parts.
%Y A106356 The k-th composition in standard-order has A124767(k) maximal runs and A333381(k) maximal anti-runs.
%Y A106356 The version for ascents/descents is A238343.
%Y A106356 The version for weak ascents/descents is A333213.
%Y A106356 Cf. A064113, A066099, A233564, A333214, A333216.
%K A106356 nonn,tabl
%O A106356 1,4
%A A106356 _Christian G. Bower_, Apr 29 2005