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 A287213 Number T(n,k) of set partitions of [n] such that the maximal absolute difference between consecutive elements within a block equals k; triangle T(n,k), n>=0, 0<=k<=max(n-1,0), read by rows. 15
 1, 1, 1, 1, 1, 3, 1, 1, 7, 5, 2, 1, 15, 18, 13, 5, 1, 31, 57, 61, 38, 15, 1, 63, 169, 248, 215, 129, 52, 1, 127, 482, 935, 1061, 836, 495, 203, 1, 255, 1341, 3368, 4835, 4789, 3573, 2108, 877, 1, 511, 3669, 11777, 20973, 25430, 22986, 16657, 9831, 4140 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS The maximal absolute difference is assumed to be zero if there is no block with consecutive elements. 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..23, flattened Wikipedia, Partition of a set FORMULA T(n,k) = A287214(n,k) - A287214(n,k-1) for k>0, T(n,0) = 1. EXAMPLE T(4,0) = 1: 1|2|3|4. T(4,1) = 7: 1234, 123|4, 12|34, 12|3|4, 1|234, 1|23|4, 1|2|34. T(4,2) = 5: 124|3, 134|2, 13|24, 13|2|4, 1|24|3. T(4,3) = 2: 14|23, 14|2|3. Triangle T(n,k) begins:   1;   1;   1,   1;   1,   3,   1;   1,   7,   5,   2;   1,  15,  18,  13,    5;   1,  31,  57,  61,   38,  15;   1,  63, 169, 248,  215, 129,  52;   1, 127, 482, 935, 1061, 836, 495, 203; MAPLE b:= proc(n, k, l) option remember; `if`(n=0, 1, b(n-1, k, map(x->       `if`(x-n>=k, [][], x), [l[], n]))+add(b(n-1, k, sort(map(x->       `if`(x-n>=k, [][], x), subsop(j=n, l)))), j=1..nops(l)))     end: A:= (n, k)-> b(n, min(k, n-1), []): 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[0, _, _] = 1; b[n_, k_, l_] := b[n, k, l] =b[n - 1, k, If[# - n >= k, Nothing, #]& /@ Append[l, n]] + Sum[b[n - 1, k, Sort[If[# - n >= k, Nothing, #]& /@ ReplacePart[l, j -> n]]], {j, 1, Length[l]}]; A[n_, k_] := b[n, Min[k, n - 1], {}]; T[n_, k_] :=  A[n, k] - If[k == 0, 0, A[n, k - 1]]; Table[Table[T[n, k], {k, 0, Max[n - 1, 0]}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *) CROSSREFS Columns k=0-10 give: A000012, A000225(n-1), A258109, A294052, A294053, A294054, A294055, A294056, A294057, A294058, A294059. Row sums and T(n+2,n+1) give A000110. T(2n,n) gives A294024. Cf. A287214, A287215, A287416, A287640. Sequence in context: A343237 A094507 A065625 * A284631 A154341 A348863 Adjacent sequences:  A287210 A287211 A287212 * A287214 A287215 A287216 KEYWORD nonn,tabf AUTHOR Alois P. Heinz, May 21 2017 STATUS approved

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Last modified December 1 22:12 EST 2021. Contains 349435 sequences. (Running on oeis4.)