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A214269
Number T(n,k) of compositions of n where the difference between largest and smallest parts equals k and adjacent parts are unequal; triangle T(n,k), n>=1, 0<=k<n, read by rows.
18
1, 1, 0, 1, 2, 0, 1, 1, 2, 0, 1, 3, 1, 2, 0, 1, 2, 8, 1, 2, 0, 1, 4, 7, 8, 1, 2, 0, 1, 2, 13, 12, 8, 1, 2, 0, 1, 4, 25, 18, 12, 8, 1, 2, 0, 1, 4, 27, 46, 23, 12, 8, 1, 2, 0, 1, 4, 43, 69, 51, 23, 12, 8, 1, 2, 0, 1, 3, 71, 111, 90, 56, 23, 12, 8, 1, 2, 0
OFFSET
1,5
LINKS
FORMULA
T(n,0) = 1, T(n,k) = A214268(n,k) - A214268(n,k-1) for k>0.
EXAMPLE
T(7,0) = 1: [7].
T(7,1) = 4: [4,3], [3,4], [2,3,2], [1,2,1,2,1].
T(7,2) = 7: [3,1,3], [3,1,2,1], [2,1,3,1], [1,3,2,1], [1,3,1,2], [1,2,3,1], [1,2,1,3].
T(7,3) = 8: [5,2], [4,2,1], [4,1,2], [2,5], [2,4,1], [2,1,4], [1,4,2], [1,2,4].
T(7,4) = 1: [1,5,1].
T(7,5) = 2: [6,1], [1,6].
Triangle T(n,k) begins:
1;
1, 0;
1, 2, 0;
1, 1, 2, 0;
1, 3, 1, 2, 0;
1, 2, 8, 1, 2, 0;
1, 4, 7, 8, 1, 2, 0;
1, 2, 13, 12, 8, 1, 2, 0;
MAPLE
b:= proc(n, k, s, t, l) option remember;
`if`(n<0, 0, `if`(n=0, 1, add(`if`(j=l, 0, b(n-j, k,
min(s, j), max(t, j), j)), j=max(1, t-k+1)..s+k-1)))
end:
A:= proc(n, k) option remember;
`if`(n=0, 1, add(b(n-j, k+1, j, j, j), j=1..n))
end:
T:= (n, k)-> A(n, k) -`if`(k=0, 0, A(n, k-1)):
seq(seq(T(n, k), k=0..n-1), n=1..14);
MATHEMATICA
b[n_, k_, s_, t_, l_] := b[n, k, s, t, l] = If[n < 0, 0, If[n == 0, 1, Sum [If[j == l, 0, b[n-j, k, Min[s, j], Max[t, j], j]], {j, Max[1, t-k+1], s+k-1}] ] ]; a[n_, k_] := a[n, k] = If[n == 0, 1, Sum[b[n - j, k+1, j, j, j], {j, 1, n}]]; t[n_, k_] := a[n, k] - If[k == 0, 0, a[n, k-1]]; Table[Table[t[n, k], {k, 0, n-1}], {n, 1, 12}] // Flatten (* Jean-François Alcover, Dec 11 2013, translated from Maple *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Alois P. Heinz, Jul 09 2012
STATUS
approved