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A262191
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Number T(n,k) of compositions of n such that k is the maximal distance between two identical parts; triangle T(n,k), n>=2, 1<=k<=n-1, read by rows.
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8
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1, 0, 1, 3, 1, 1, 4, 4, 2, 1, 5, 6, 6, 3, 1, 12, 13, 12, 9, 4, 1, 21, 23, 25, 21, 13, 5, 1, 36, 42, 46, 46, 34, 18, 6, 1, 43, 68, 88, 92, 80, 52, 24, 7, 1, 88, 119, 152, 180, 172, 132, 76, 31, 8, 1, 133, 197, 267, 330, 352, 304, 208, 107, 39, 9, 1
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table;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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2,4
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LINKS
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EXAMPLE
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T(6,1) = 5: 33, 114, 411, 1122, 2211.
T(6,2) = 6: 141, 222, 1113, 1212, 2121, 3111.
T(6,3) = 6: 1131, 1221, 1311, 2112, 11112, 21111.
T(6,4) = 3: 11121, 11211, 12111.
T(6,5) = 1: 111111.
Triangle T(n,k) begins:
n\k: 1 2 3 4 5 6 7 8 9 10 11
---+----------------------------------------------------
02 : 1;
03 : 0, 1;
04 : 3, 1, 1;
05 : 4, 4, 2, 1;
06 : 5, 6, 6, 3, 1;
07 : 12, 13, 12, 9, 4, 1;
08 : 21, 23, 25, 21, 13, 5, 1;
09 : 36, 42, 46, 46, 34, 18, 6, 1;
10 : 43, 68, 88, 92, 80, 52, 24, 7, 1;
11 : 88, 119, 152, 180, 172, 132, 76, 31, 8, 1;
12 : 133, 197, 267, 330, 352, 304, 208, 107, 39, 9, 1;
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MAPLE
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b:= proc(n, s, l) option remember; `if`(n=0, 1, add(
`if`(j in s, 0, b(n-j, s union {`if`(l=[], j, l[1])},
`if`(l=[], [], [subsop(1=NULL, l)[], j]))), j=1..n))
end:
T:= (n, k)-> b(n, {}, [0$k]) -b(n, {}, [0$(k-1)]):
seq(seq(T(n, k), k=1..n-1), n=2..14);
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MATHEMATICA
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b[n_, s_, l_] := b[n, s, l] = If[n==0, 1, Sum[If[MemberQ[s, j], 0, b[n-j, s ~Union~ {If[l=={}, j, l[[1]]]}, If[l=={}, {}, Append[Rest[l], j]]]], {j, 1, n}]]; T[n_, k_] := b[n, {}, Array[0&, k]] - b[n, {}, Array[0&, k-1]]; Table[T[n, k], {n, 2, 14}, { k, 1, n-1}] // Flatten (* Jean-François Alcover, Feb 08 2017, translated from Maple *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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