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A287640 Number T(n,k) of set partitions of [n], where k is minimal such that for all j in [n]: j is member of block b implies b = 1 or at least one of j-1, ..., j-k is member of a block >= b-1; triangle T(n,k), n >= 0, 0 <= k <= max(floor(n/2), n-2), read by rows.

%I A287640 #25 Jan 06 2022 14:28:26
%S A287640 1,1,1,1,1,4,1,13,1,1,41,9,1,1,131,59,11,1,1,428,344,88,15,1,1,1429,
%T A287640 1906,634,146,23,1,1,4861,10345,4389,1231,280,39,1,1,16795,55901,
%U A287640 30006,9835,2763,602,71,1,1,58785,303661,205420,77178,25014,6967,1408,135,1
%N A287640 Number T(n,k) of set partitions of [n], where k is minimal such that for all j in [n]: j is member of block b implies b = 1 or at least one of j-1, ..., j-k is member of a block >= b-1; triangle T(n,k), n >= 0, 0 <= k <= max(floor(n/2), n-2), read by rows.
%H A287640 Alois P. Heinz, <a href="/A287640/b287640.txt">Rows n = 0..20, flattened</a>
%H A287640 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A287640 T(n,k) = A287641(n,k) - A287641(n,k-1) for k>0, T(n,0) = 1.
%F A287640 T(n+4,n+1) = A168415(n) for n>0.
%e A287640 T(4,0) = 1: 1234.
%e A287640 T(4,1) = 13: 123|4, 124|3, 12|34, 12|3|4, 134|2, 13|24, 14|23, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34, 1|2|3|4.
%e A287640 T(4,2) = 1: 13|2|4.
%e A287640 T(5,2) = 9: 124|3|5, 135|2|4, 13|25|4, 13|2|45, 13|2|4|5, 14|23|5, 14|2|35, 14|2|3|5, 1|24|3|5.
%e A287640 T(6,3) = 11: 1245|3|6, 1346|2|5, 134|26|5, 134|2|56, 134|2|5|6, 145|23|6, 145|2|36, 145|2|3|6, 14|25|3|6, 15|24|3|6, 1|245|3|6.
%e A287640 T(6,4) = 1: 1345|2|6.
%e A287640 T(7,4) = 15: 12456|3|7, 13457|2|6, 1345|27|6, 1345|2|67, 1345|2|6|7, 1456|23|7, 1456|2|37, 1456|2|3|7, 145|26|3|7, 146|25|3|7, 14|256|3|7, 156|24|3|7, 15|246|3|7, 16|245|3|7, 1|2456|3|7.
%e A287640 Triangle T(n,k) begins:
%e A287640   1;
%e A287640   1;
%e A287640   1,    1;
%e A287640   1,    4;
%e A287640   1,   13,     1;
%e A287640   1,   41,     9,    1;
%e A287640   1,  131,    59,   11,    1;
%e A287640   1,  428,   344,   88,   15,   1;
%e A287640   1, 1429,  1906,  634,  146,  23,  1;
%e A287640   1, 4861, 10345, 4389, 1231, 280, 39, 1;
%e A287640   ...
%p A287640 b:= proc(n, l) option remember; `if`(n=0 or l=[], 1, add(b(n-1,
%p A287640       [seq(max(l[i], j), i=2..nops(l)), j]), j=1..l[1]+1))
%p A287640     end:
%p A287640 T:= (n, k)-> `if`(k=0, 1, b(n, [0$k])-b(n, [0$k-1])):
%p A287640 seq(seq(T(n, k), k=0..max(n/2, n-2)), n=0..12);
%t A287640 b[n_, l_] := b[n, l] = If[n == 0 || l == {}, 1, Sum[b[n-1, Append[Table[ Max[l[[i]], j], {i, 2, Length[l]}], j]], {j, 1, l[[1]] + 1}]];
%t A287640 T[n_, k_] := If[k == 0, 1, b[n, Table[0, k]] - b[n, Table[0, k - 1]]];
%t A287640 Table[T[n, k], {n, 0, 12}, { k, 0, Max[n/2, n - 2]}] // Flatten (* _Jean-François Alcover_, May 22 2018, translated from Maple *)
%Y A287640 Columns k=0-1 give: A000012, A001453.
%Y A287640 Row sums give A000110.
%Y A287640 Cf. A168415, A287213, A287215, A287416, A287641.
%K A287640 nonn,tabf
%O A287640 0,6
%A A287640 _Alois P. Heinz_, May 28 2017