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

%I #5 May 02 2016 12:33:44

%S 1,1,2,4,9,23,66,204,664,2273,8283,32463,136434,605848

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

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

%e a(3) = 4: 123, 12|3, 13|2, 1|23.

%e a(4) = 9: 1234, 123|4, 124|3, 12|34, 134|2, 13|24, 14|23, 1|234, 14|2|3.

%e a(5) = 23: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 125|3|4, 1345|2, 134|25, 135|24, 13|245, 13|25|4, 145|23, 14|235, 15|234, 1|2345, 15|23|4, 145|2|3, 14|25|3, 14|2|35, 15|2|34.

%e a(6) = 66: 123456, 12345|6, 12346|5, 1234|56, 12356|4, 1235|46, 1236|45, 123|456, 1236|4|5, 12456|3, 1245|36, 1246|35, 124|356, 124|36|5, 1256|34, 125|346, 126|345, 12|3456, 126|34|5, 1256|3|4, 125|36|4, 125|3|46, 126|3|45, 13456|2, 1345|26, 1346|25, 134|256, 134|26|5, 1356|24, 135|246, 136|245, 13|2456, 136|24|5, 136|25|4, 13|256|4, 13|25|46, 13|26|45, 1456|23, 145|236, 146|235, 14|2356, 14|236|5, 156|234, 15|2346, 16|2345, 1|23456, 16|234|5, 156|23|4, 15|236|4, 15|23|46, 16|23|45, 1456|2|3, 145|26|3, 145|2|36, 146|25|3, 14|256|3, 14|25|36, 146|2|35, 14|26|35, 14|2|356, 15|24|36, 16|24|35, 156|2|34, 15|26|34, 15|2|346, 16|2|345.

%Y Cf. A000110, A185982, A271270, A272064, A272105.

%K nonn,more

%O 0,3

%A _Alois P. Heinz_, Apr 25 2016