A271270 - OEIS

%I #17 Jan 30 2017 09:09:14

%S 1,1,2,5,14,43,145,536,2157,9371,43630,216397,1137703,6313675,

%T 36848992,225464838,1442216870,9620746697,66781675113,481413175433,

%U 3597627996006,27825925290597,222422033403527,1834910286704787,15603508329713182,136616625732498989

%N Number of set partitions of [n] such that for each pair of consecutive blocks (b,b+1) at least one pair of consecutive numbers (i,i+1) exists with i member of b and i+1 member of b+1.

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>

%F a(n) = A000110(n) - A271271(n).

%e A000110(4) - a(4) = 15 - 14 = 1: 13|2|4.

%e A000110(5) - a(5) = 52 - 43 = 9: 124|3|5, 134|2|5, 135|2|4, 13|25|4, 13|2|45, 13|2|4|5, 14|23|5, 14|2|3|5, 1|24|3|5.

%p b:= proc(n, i, m, l) option remember; `if`(n=0,

%p       `if`({l[], 1}={1}, 1, 0), add(b(n-1, j, max(m, j),

%p       `if`(j=m+1, [l[], `if`(j=i+1, 1, 0)],

%p       `if`(j=i+1, subsop(j=1, l), l))), j=1..m+1))

%p     end:

%p a:= n-> b(n, 0$2, []):

%p seq(a(n), n=0..18);

%t b[n_, i_, m_, l_] := b[n, i, m, l] = If[n == 0, If[Union[l, {1}] == {1}, 1, 0], Sum[b[n-1, j, Max[m, j], If[j == m+1, Join[l, If[j == i+1, {1}, {0}] ], If[j == i+1, ReplacePart[l, j -> 1], l]]], {j, 1, m+1}]]; a[n_] := b[n, 0, 0, {}]; Table[a[n], {n, 0, 18}] (* _Jean-François Alcover_, Jan 30 2017, translated from Maple *)

%Y Cf. A000110, A185982, A271271, A271272, A272064.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Apr 03 2016