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 A287215 Number T(n,k) of set partitions of [n] such that the maximal absolute difference between the least elements of consecutive blocks equals k; triangle T(n,k), n>=0, 0<=k<=max(n-1,0), read by rows. 18
 1, 1, 1, 1, 1, 3, 1, 1, 8, 5, 1, 1, 22, 21, 7, 1, 1, 65, 86, 39, 11, 1, 1, 209, 361, 209, 77, 19, 1, 1, 732, 1584, 1123, 493, 171, 35, 1, 1, 2780, 7315, 6153, 3124, 1293, 413, 67, 1, 1, 11377, 35635, 34723, 20019, 9320, 3709, 1059, 131, 1, 1, 49863, 183080, 202852, 130916, 66992, 30396, 11373, 2837, 259, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS The maximal absolute difference is assumed to be zero if there are fewer than two blocks. T(n,k) is defined for all n,k >= 0.  The triangle contains only the positive terms. T(n,k) = 0 if k>=n and k>0. LINKS Alois P. Heinz, Rows n = 0..141, flattened Wikipedia, Partition of a set FORMULA T(n,k) = A287216(n,k) - A287216(n,k-1) for k>0, T(n,0) = 1. EXAMPLE T(4,0) = 1: 1234. T(4,1) = 8: 134|2, 13|24, 14|23, 1|234, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4. T(4,2) = 5: 124|3, 12|34, 12|3|4, 13|2|4, 1|23|4. T(4,3) = 1: 123|4. Triangle T(n,k) begins:   1;   1;   1,   1;   1,   3,    1;   1,   8,    5,    1;   1,  22,   21,    7,   1;   1,  65,   86,   39,  11,   1;   1, 209,  361,  209,  77,  19,  1;   1, 732, 1584, 1123, 493, 171, 35, 1; MAPLE b:= proc(n, k, m, l) option remember; `if`(n<1, 1,      `if`(l-n>k, 0, b(n-1, k, m+1, n))+m*b(n-1, k, m, l))     end: A:= (n, k)-> b(n-1, min(k, n-1), 1, n): T:= (n, k)-> A(n, k)-`if`(k=0, 0, A(n, k-1)): seq(seq(T(n, k), k=0..max(n-1, 0)), n=0..12); MATHEMATICA b[n_, k_, m_, l_] := b[n, k, m, l] = If[n < 1, 1, If[l - n > k, 0, b[n - 1, k, m + 1, n]] + m*b[n - 1, k, m, l]]; A[n_, k_] := b[n - 1, Min[k, n - 1], 1, n]; T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]]; Table[T[n, k], {n, 0, 12}, {k, 0, Max[n - 1, 0]}] // Flatten (* Jean-François Alcover, May 19 2018, after Alois P. Heinz *) CROSSREFS Columns k=0-10 give: A000012, A003101(n-1), A322875, A322876, A322877, A322878, A322879, A322880, A322881, A322882, A322883. Row sums give A000110. T(2n,n) gives A322884. Cf. A287213, A287216, A287416, A287640. Sequence in context: A098747 A122897 A117425 * A168216 A091698 A134380 Adjacent sequences:  A287212 A287213 A287214 * A287216 A287217 A287218 KEYWORD nonn,tabf AUTHOR Alois P. Heinz, May 21 2017 STATUS approved

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Last modified May 21 06:36 EDT 2022. Contains 353889 sequences. (Running on oeis4.)