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%I #25 Oct 18 2018 15:42:16
%S 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,
%T 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,
%U 15,52,202,780,2317,2781,1,1,1,2,5,15,52,203,857,3440,10096,11378,1
%N 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.
%H Alois P. Heinz, <a href="/A287216/b287216.txt">Antidiagonals n = 0..140, flattened</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A(n,k) = Sum_{j=0..k} A287215(n,j).
%e A(4,0) = 1: 1234.
%e 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.
%e 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.
%e 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.
%e Square array A(n,k) begins:
%e 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 1, 1, 1, 1, 1, 1, ...
%e 1, 2, 2, 2, 2, 2, 2, 2, ...
%e 1, 4, 5, 5, 5, 5, 5, 5, ...
%e 1, 9, 14, 15, 15, 15, 15, 15, ...
%e 1, 23, 44, 51, 52, 52, 52, 52, ...
%e 1, 66, 152, 191, 202, 203, 203, 203, ...
%e 1, 210, 571, 780, 857, 876, 877, 877, ...
%p b:= proc(n, k, m, l) option remember; `if`(n<1, 1,
%p `if`(l-n>k, 0, b(n-1, k, m+1, n))+m*b(n-1, k, m, l))
%p end:
%p A:= (n, k)-> b(n-1, min(k, n-1), 1, n):
%p seq(seq(A(n, d-n), n=0..d), d=0..12);
%t 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]];
%t A[n_, k_] := b[n - 1, Min[k, n - 1], 1, n];
%t Table[A[n, d - n], {d, 0, 12}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Apr 30 2018, after _Alois P. Heinz_ *)
%Y Columns k=0-10 give: A000012, A026898(n-1) for n>0, A287252, A287253, A287254, A287255, A287256, A287257, A287258, A287259, A287260.
%Y Main diagonal gives A000110.
%Y Cf. A287214, A287215, A287417, A287641.
%K nonn,tabl
%O 0,9
%A _Alois P. Heinz_, May 21 2017
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