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A362893 Number of partitions of [n] whose blocks can be ordered such that the i-th block has at least i elements and no block j > i has an element smaller than the i-th smallest element of block i.

%I A362893 #13 May 09 2023 20:16:06
%S A362893 1,1,1,2,5,12,28,69,193,614,2103,7359,25660,88914,309502,1102146,
%T A362893 4092840,16046224,66410789,286905421,1273646720,5729762139,
%U A362893 25881820352,116872997038,527375160184,2384407416357,10856086444051,50097994816979,235937202788389
%N A362893 Number of partitions of [n] whose blocks can be ordered such that the i-th block has at least i elements and no block j > i has an element smaller than the i-th smallest element of block i.
%H A362893 Alois P. Heinz, <a href="/A362893/b362893.txt">Table of n, a(n) for n = 0..890</a>
%H A362893 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A362893 a(0) = 1: (), the empty partition.
%e A362893 a(1) = 1: 1.
%e A362893 a(2) = 1: 12.
%e A362893 a(3) = 2: 123, 1|23.
%e A362893 a(4) = 5: 1234, 12|34, 13|24, 14|23, 1|234.
%e A362893 a(5) = 12: 12345, 123|45, 124|35, 125|34, 12|345, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234, 1|2345.
%e A362893 a(6) = 28: 123456, 1234|56, 1235|46, 1236|45, 123|456, 1245|36, 1246|35, 124|356, 1256|34, 125|346, 126|345, 12|3456, 1345|26, 1346|25, 134|256, 1356|24, 135|246, 136|245, 13|2456, 1456|23, 145|236, 146|235, 14|2356, 156|234, 15|2346, 16|2345, 1|23456, 1|23|456.
%e A362893 a(7) = 69: 1234567, 12345|67, 12346|57, 12347|56, 1234|567, 12356|47, 12357|46, 1235|467, 12367|45, 1236|457, 1237|456, 123|4567, 12456|37, 12457|36, 1245|367, 12467|35, 1246|357, 1247|356, 124|3567, 12567|34, 1256|347, 1257|346, 125|3467, 1267|345, 126|3457, 127|3456, 12|34567, 12|34|567, 13456|27, 13457|26, 1345|267, 13467|25, 1346|257, 1347|256, 134|2567, 13567|24, 1356|247, 1357|246, 135|2467, 1367|245, 136|2457, 137|2456, 13|24567, 13|24|567, 14567|23, 1456|237, 1457|236, 145|2367, 1467|235, 146|2357, 147|2356, 14|23567, 14|23|567, 1567|234, 156|2347, 157|2346, 15|23467, 167|2345, 16|23457, 17|23456, 1|234567, 1|234|567, 15|23|467, 1|235|467, 16|23|457, 1|236|457, 17|23|456, 1|237|456, 1|23|4567.
%p A362893 b:= proc(n, t) option remember; `if`(n=0 or n=t, 1,
%p A362893       add(b(n-j, t+1)*binomial(n-t, j-t), j=t..n))
%p A362893     end:
%p A362893 a:= n-> b(n, 1):
%p A362893 seq(a(n), n=0..28);
%Y A362893 Cf. A000110, A362549, A362635.
%K A362893 nonn
%O A362893 0,4
%A A362893 _Alois P. Heinz_, May 08 2023