A270954 - OEIS

%I #10 May 27 2018 07:00:20

%S 1,1,2,4,9,25,84,323,1377,6412,32312,174941,1011357,6210298,40323011,

%T 275763910,1979709852,14875239212,116679269248,953201694216,

%U 8093501305721,71291395351760,650357174742217,6134966541625355,59759476024690454,600309156303711764

%N Number of set partitions of [n] having no pairs (m,m+1) such that m+1 is in some block b and m is in block b+1.

%H Alois P. Heinz, <a href="/A270954/b270954.txt">Table of n, a(n) for n = 0..300</a>

%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, 1|23, 1|2|3.

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

%p b:= proc(n, i, m) option remember; `if`(n=0, 1, add(

%p      `if`(j=i-1, 0, b(n-1, j, max(m, j))), j=1..m+1))

%p     end:

%p a:= n-> b(n, 1, 0):

%p seq(a(n), n=0..25);

%t b[n_, i_, m_] := b[n, i, m] = If[n == 0, 1, Sum[If[j == i - 1, 0, b[n - 1, j, Max[m, j]]], {j, 1, m + 1}]];

%t a[n_] := b[n, 1, 0];

%t Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, May 27 2018, translated from Maple *)

%Y Column k=0 of A270953.

%K nonn

%O 0,3

%A _Alois P. Heinz_, Mar 26 2016