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 A276727 Number T(n,k) of set partitions of [n] where k is minimal such that for each block b the smallest integer interval containing b has at most k elements; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 13
 1, 0, 1, 0, 1, 1, 0, 1, 2, 2, 0, 1, 4, 5, 5, 0, 1, 7, 12, 17, 15, 0, 1, 12, 29, 45, 64, 52, 0, 1, 20, 66, 121, 201, 265, 203, 0, 1, 33, 145, 336, 585, 966, 1197, 877, 0, 1, 54, 315, 901, 1741, 3172, 4971, 5852, 4140, 0, 1, 88, 676, 2347, 5375, 10100, 18223, 27267, 30751, 21147 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 LINKS Alois P. Heinz, Rows n = 0..20, flattened FORMULA T(n,k) = A276719(n,k) - A276719(n,k-1) for k>0, T(n,0) = A000007(n). EXAMPLE T(4,1) = 1: 1|2|3|4. T(4,2) = 4: 12|34, 12|3|4, 1|23|4, 1|2|34. T(4,3) = 5: 123|4, 13|24, 13|2|4, 1|234, 1|24|3. T(4,4) = 5: 1234, 124|3, 134|2, 14|23, 14|2|3. T(5,4) = 17: 1234|5, 124|35, 124|3|5, 134|25, 134|2|5, 13|245, 13|25|4, 14|235, 14|23|5, 1|2345, 1|235|4, 14|25|3, 14|2|35, 14|2|3|5, 1|245|3, 1|25|34, 1|25|3|4. Triangle T(n,k) begins:   1;   0, 1;   0, 1,  1;   0, 1,  2,   2;   0, 1,  4,   5,   5;   0, 1,  7,  12,  17,  15;   0, 1, 12,  29,  45,  64,  52;   0, 1, 20,  66, 121, 201, 265,  203;   0, 1, 33, 145, 336, 585, 966, 1197, 877; MAPLE b:= proc(n, m, l) option remember; `if`(n=0, 1,       add(b(n-1, max(m, j), [subsop(1=NULL, l)[],       `if`(j<=m, 0, j)]), j={l[], m+1} minus {0}))     end: A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, b(n, 0, [0\$(k-1)]))): T:= (n, k)-> A(n, k) -`if`(k=0, 0, A(n, k-1)): seq(seq(T(n, k), k=0..n), n=0..12); MATHEMATICA b[n_, m_, l_List] := b[n, m, l] = If[n == 0, 1, Sum[b[n - 1, Max[m, j], Append[ReplacePart[l, 1 -> Nothing], If[j <= m, 0, j]]], {j, Append[l, m + 1] ~Complement~ {0}}]]; A[n_, k_] := If[n == 0, 1, If[k < 2, k, b[n, 0, Array[0&, k - 1]]]]; T [n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]]; Table[T[n, k], {n, 0, 12}, { k, 0, n}] // Flatten (* Jean-François Alcover, Feb 04 2017, translated from Maple *) CROSSREFS Columns k=0-10 give: A000007, A057427, A000071(n+1), A320553, A320554, A320555, A320556, A320557, A320558, A320559, A320560. Row sums give A000110. Main diagonal gives A000110(n-1) for n>0. T(2n,n) gives A276728. Cf. A263757, A276719, A276891. Sequence in context: A317575 A295653 A146326 * A267617 A158852 A188285 Adjacent sequences:  A276724 A276725 A276726 * A276728 A276729 A276730 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, Sep 16 2016 STATUS approved

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Last modified May 22 11:02 EDT 2022. Contains 353949 sequences. (Running on oeis4.)