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A106356
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Triangle T(n,k) 0<=k<n : Number of compositions of n with k adjacent equal parts.
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94
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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
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OFFSET
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1,4
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COMMENTS
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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
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LINKS
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A. Knopfmacher and H. Prodinger, On Carlitz compositions, European Journal of Combinatorics, Vol. 19, 1998, pp. 579-589.
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EXAMPLE
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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;
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)
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MAPLE
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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):
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MATHEMATICA
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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 *)
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CROSSREFS
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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.
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KEYWORD
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AUTHOR
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STATUS
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approved
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