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A287641 Number A(n,k) of set partitions of [n] such that j is member of block b only if b = 1 or at least one of j-1, ..., j-k is member of a block >= b-1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

%I A287641 #26 Oct 18 2018 15:43:51
%S A287641 1,1,1,1,1,1,1,1,2,1,1,1,2,5,1,1,1,2,5,14,1,1,1,2,5,15,42,1,1,1,2,5,
%T A287641 15,51,132,1,1,1,2,5,15,52,191,429,1,1,1,2,5,15,52,202,773,1430,1,1,1,
%U A287641 2,5,15,52,203,861,3336,4862,1,1,1,2,5,15,52,203,876,3970,15207,16796,1
%N A287641 Number A(n,k) of set partitions of [n] such that j is member of block b only if b = 1 or at least one of j-1, ..., j-k is member of a block >= b-1; square array A(n,k), n>=0, k>=0, read by antidiagonals.
%H A287641 Alois P. Heinz, <a href="/A287641/b287641.txt">Antidiagonals n = 0..34, flattened</a>
%H A287641 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A287641 A(n,k) = Sum_{j=0..k} A287640(n,j).
%e A287641 A(5,0) = 1: 12345.
%e A287641 A(5,1) = 42 = 52 - 10 = A000110(5) - 10 counts all set partitions of [5] except: 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, 134|2|5.
%e A287641 A(5,2) = 51 = 52 - 1 = A000110(5) - 1 counts all set partitions of [5] except: 134|2|5.
%e A287641 Square array A(n,k) begins:
%e A287641   1,   1,   1,   1,   1,   1,   1,   1, ...
%e A287641   1,   1,   1,   1,   1,   1,   1,   1, ...
%e A287641   1,   2,   2,   2,   2,   2,   2,   2, ...
%e A287641   1,   5,   5,   5,   5,   5,   5,   5, ...
%e A287641   1,  14,  15,  15,  15,  15,  15,  15, ...
%e A287641   1,  42,  51,  52,  52,  52,  52,  52, ...
%e A287641   1, 132, 191, 202, 203, 203, 203, 203, ...
%e A287641   1, 429, 773, 861, 876, 877, 877, 877, ...
%p A287641 b:= proc(n, l) option remember; `if`(n=0, 1, add(b(n-1,
%p A287641       [seq(max(l[i], j), i=2..nops(l)), j]), j=1..l[1]+1))
%p A287641     end:
%p A287641 A:= (n, k)-> `if`(k=0, 1, b(n, [0$k])):
%p A287641 seq(seq(A(n, d-n), n=0..d), d=0..12);
%t A287641 b[0, _] = 1; b[n_, l_List] := b[n, l] = Sum[b[n - 1, Append[ Table[ Max[ l[[i]], j], {i, 2, Length[l]}], j]], {j, 1, l[[1]] + 1}];
%t A287641 A[n_, k_] := If[k == 0, 1, b[n, Table[0, k]]];
%t A287641 Table[A[n, d - n], {d, 0, 12}, {n, 0, d}] // Flatten (* _Jean-François Alcover_, Apr 30 2018, after _Alois P. Heinz_ *)
%Y A287641 Columns k=0-10 give: A000012, A000108, A275605, A287666, A287667, A287668, A287669, A287670, A287671, A287672, A287673.
%Y A287641 Main diagonal gives A000110.
%Y A287641 Cf. A064645, A287214, A287216, A287417, A287640.
%K A287641 nonn,tabl
%O A287641 0,9
%A A287641 _Alois P. Heinz_, May 28 2017