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
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
Examples
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;
Links
- Alois P. Heinz, Rows n = 1..150, flattened
Crossrefs
Programs
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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 *)