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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.

%I A287215 #25 Dec 29 2018 09:36:51
%S A287215 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,
%T A287215 19,1,1,732,1584,1123,493,171,35,1,1,2780,7315,6153,3124,1293,413,67,
%U A287215 1,1,11377,35635,34723,20019,9320,3709,1059,131,1,1,49863,183080,202852,130916,66992,30396,11373,2837,259,1
%N 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.
%C A287215 The maximal absolute difference is assumed to be zero if there are fewer than two blocks.
%C A287215 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.
%H A287215 Alois P. Heinz, <a href="/A287215/b287215.txt">Rows n = 0..141, flattened</a>
%H A287215 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A287215 T(n,k) = A287216(n,k) - A287216(n,k-1) for k>0, T(n,0) = 1.
%e A287215 T(4,0) = 1: 1234.
%e A287215 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.
%e A287215 T(4,2) = 5: 124|3, 12|34, 12|3|4, 13|2|4, 1|23|4.
%e A287215 T(4,3) = 1: 123|4.
%e A287215 Triangle T(n,k) begins:
%e A287215   1;
%e A287215   1;
%e A287215   1,   1;
%e A287215   1,   3,    1;
%e A287215   1,   8,    5,    1;
%e A287215   1,  22,   21,    7,   1;
%e A287215   1,  65,   86,   39,  11,   1;
%e A287215   1, 209,  361,  209,  77,  19,  1;
%e A287215   1, 732, 1584, 1123, 493, 171, 35, 1;
%p A287215 b:= proc(n, k, m, l) option remember; `if`(n<1, 1,
%p A287215      `if`(l-n>k, 0, b(n-1, k, m+1, n))+m*b(n-1, k, m, l))
%p A287215     end:
%p A287215 A:= (n, k)-> b(n-1, min(k, n-1), 1, n):
%p A287215 T:= (n, k)-> A(n, k)-`if`(k=0, 0, A(n, k-1)):
%p A287215 seq(seq(T(n, k), k=0..max(n-1, 0)), n=0..12);
%t A287215 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 A287215 A[n_, k_] := b[n - 1, Min[k, n - 1], 1, n];
%t A287215 T[n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]];
%t A287215 Table[T[n, k], {n, 0, 12}, {k, 0, Max[n - 1, 0]}] // Flatten (* _Jean-François Alcover_, May 19 2018, after _Alois P. Heinz_ *)
%Y A287215 Columns k=0-10 give: A000012, A003101(n-1), A322875, A322876, A322877, A322878, A322879, A322880, A322881, A322882, A322883.
%Y A287215 Row sums give A000110.
%Y A287215 T(2n,n) gives A322884.
%Y A287215 Cf. A287213, A287216, A287416, A287640.
%K A287215 nonn,tabf
%O A287215 0,6
%A A287215 _Alois P. Heinz_, May 21 2017