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A271272 Number of set partitions of [n] into m blocks such that for each pair of distinct cyclically consecutive blocks (b,c) = (b,(b mod m)+1) at least one pair of numbers (i,j) = (i,(i mod n)+1) exists with i member of b and j member of c.

%I A271272 #9 Feb 15 2017 12:01:07
%S A271272 1,1,2,5,13,36,110,374,1404,5750,25419,120325,606210,3234618,18202851,
%T A271272 107647893,666903189,4316424771,29116689197,204259773724,
%U A271272 1487336089532,11221857590608,87591879539120,706286859093554,5875489876724901,50364717424939105,444367708336858660
%N A271272 Number of set partitions of [n] into m blocks such that for each pair of distinct cyclically consecutive blocks (b,c) = (b,(b mod m)+1) at least one pair of numbers (i,j) = (i,(i mod n)+1) exists with i member of b and j member of c.
%H A271272 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A271272 a(n) = A000110(n) - A271273(n).
%e A271272 A000110(4) - a(4) = 15 - 13 = 2: 13|2|4, 1|24|3.
%e A271272 A000110(5) - a(5) = 52 - 36 = 16: 124|3|5, 12|35|4, 134|2|5, 135|2|4, 13|25|4, 13|2|45, 13|2|4|5, 14|23|5, 1|235|4, 14|2|3|5, 15|24|3, 1|245|3, 1|24|3|5, 1|25|34, 1|25|3|4, 1|2|35|4.
%p A271272 b:= proc(n, i, m, l) option remember; `if`(n=0,
%p A271272      `if`(l=[] or {l[]}={1} or i=m and {subsop(1=1, l)[]}=
%p A271272       {1}, 1, 0), add(b(n-1, j, max(m, j), `if`(l=[], [1],
%p A271272      `if`(j=m+1, subsop(1=0, `if`(j=i+1, [l[],1], [l[],0])),
%p A271272      `if`(j=i+1 or j=1 and i=m, subsop(j=1, l), l)))), j=1..m+1))
%p A271272     end:
%p A271272 a:= n-> b(n, 0$2, []):
%p A271272 seq(a(n), n=0..18);
%t A271272 b[n_, i_, m_, l_] := b[n, i, m, l] = If[n==0, If[l=={} || Union[l]=={1} || i==m && Union @ ReplacePart[l, 1 -> 1] == {1}, 1, 0], Sum[b[n-1, j, Max[m, j], If[l=={}, {1}, If[j==m+1, ReplacePart[If[j==i+1, Append[l, 1], Append[l, 0]], 1 -> 0], If[j==i+1 || j==1 && i==m, ReplacePart[l, j -> 1], l]]]], {j, 1, m+1}]]; a[n_] := b[n, 0, 0, {}]; Table[a[n], {n, 0, 18} ] (* _Jean-François Alcover_, Feb 15 2017, translated from Maple *)
%Y A271272 Cf. A000110, A185983, A271270, A271273.
%K A271272 nonn
%O A271272 0,3
%A A271272 _Alois P. Heinz_, Apr 03 2016