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 A287216 Number A(n,k) of set partitions of [n] such that all absolute differences between least elements of consecutive blocks are <= k; square array A(n,k), n>=0, k>=0, read by antidiagonals. 15
 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 4, 1, 1, 1, 2, 5, 9, 1, 1, 1, 2, 5, 14, 23, 1, 1, 1, 2, 5, 15, 44, 66, 1, 1, 1, 2, 5, 15, 51, 152, 210, 1, 1, 1, 2, 5, 15, 52, 191, 571, 733, 1, 1, 1, 2, 5, 15, 52, 202, 780, 2317, 2781, 1, 1, 1, 2, 5, 15, 52, 203, 857, 3440, 10096, 11378, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,9 LINKS Alois P. Heinz, Antidiagonals n = 0..140, flattened Wikipedia, Partition of a set FORMULA A(n,k) = Sum_{j=0..k} A287215(n,j). EXAMPLE A(4,0) = 1: 1234. A(4,1) = 9: 1234, 134|2, 13|24, 14|23, 1|234, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4. A(4,2) = 14: 1234, 124|3, 12|34, 12|3|4, 134|2, 13|24, 13|2|4, 14|23, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4. A(5,1) = 23: 12345, 1345|2, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234, 1|2345, 145|2|3, 14|25|3, 14|2|35, 15|24|3, 1|245|3, 1|24|35, 15|2|34, 1|25|34, 1|2|345, 15|2|3|4, 1|25|3|4, 1|2|35|4, 1|2|3|45, 1|2|3|4|5. Square array A(n,k) begins:   1,   1,   1,   1,   1,   1,   1,   1, ...   1,   1,   1,   1,   1,   1,   1,   1, ...   1,   2,   2,   2,   2,   2,   2,   2, ...   1,   4,   5,   5,   5,   5,   5,   5, ...   1,   9,  14,  15,  15,  15,  15,  15, ...   1,  23,  44,  51,  52,  52,  52,  52, ...   1,  66, 152, 191, 202, 203, 203, 203, ...   1, 210, 571, 780, 857, 876, 877, 877, ... 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): seq(seq(A(n, d-n), n=0..d), d=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]; Table[A[n, d - n], {d, 0, 12}, {n, 0, d}] // Flatten (* Jean-François Alcover, Apr 30 2018, after Alois P. Heinz *) CROSSREFS Columns k=0-10 give: A000012, A026898(n-1) for n>0, A287252, A287253, A287254, A287255, A287256, A287257, A287258, A287259, A287260. Main diagonal gives A000110. Cf. A287214, A287215, A287417, A287641. Sequence in context: A216656 A295679 A287214 * A145515 A267383 A332648 Adjacent sequences:  A287213 A287214 A287215 * A287217 A287218 A287219 KEYWORD nonn,tabl AUTHOR Alois P. Heinz, May 21 2017 STATUS approved

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Last modified January 17 06:12 EST 2022. Contains 350378 sequences. (Running on oeis4.)