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Number of partitions of [n] avoiding blocks containing their own index when blocks are ordered with decreasing largest elements.
3

%I #18 May 16 2022 04:58:41

%S 1,0,1,1,6,16,73,298,1453,7366,40689,238258,1483306,9746839,67415262,

%T 489048716,3710659737,29372630485,242021348787,2071598497189,

%U 18386889241210,168944811545046,1604584556714162,15731291424746912,159001720653174800,1654891767547439393

%N Number of partitions of [n] avoiding blocks containing their own index when blocks are ordered with decreasing largest elements.

%H Alois P. Heinz, <a href="/A350649/b350649.txt">Table of n, a(n) for n = 0..576</a>

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

%e a(4) = 6: 432|1, 42|31, 42|3|1, 4|31|2, 4|3|21, 4|3|2|1.

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

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

%p end:

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

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

%t b[n_, m_] := b[n, m] = If[n == 0, 1, Sum[

%t If[j == n, 0, b[n-1, Max[m, j]]], {j, 1, m+1}]];

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

%t Table[a[n], {n, 0, 25}] (* _Jean-François Alcover_, May 16 2022, after _Alois P. Heinz_ *)

%Y Column k=0 of A350647, A350684.

%K nonn

%O 0,5

%A _Alois P. Heinz_, Jan 09 2022