A363071 Number of partitions of [n] into m blocks that are ordered with increasing least elements and where block j contains n+1-j (m in {0..ceiling(n/2)}, j in {1..m}).

1, 1, 1, 2, 3, 6, 13, 31, 80, 222, 659, 2082, 6966, 24574, 91067, 353443, 1432909, 6054025, 26599192, 121295345, 573065538, 2800640187, 14137645933, 73619324824, 394979697320, 2180911872495, 12380240599262, 72181691321844, 431857838950302, 2649144684462775

Offset

0,4

Links

Alois P. Heinz, Table of n, a(n) for n = 0..665

Wikipedia, Partition of a set

Formula

a(n) = Sum_{j=0..ceiling(n/2)} (Stirling2(n-j,j) + Stirling2(n-j,j-1)).

a(n) = A171367(n) + A171367(n-1).

Example

a(0) = 1: (), the empty partition.
a(1) = 1: 1.
a(2) = 1: 12.
a(3) = 2: 123, 13|2.
a(4) = 3: 1234, 124|3, 14|23.
a(5) = 6: 12345, 1235|4, 125|34, 135|24, 15|234, 15|24|3.
a(6) = 13: 123456, 12346|5, 1236|45, 1246|35, 126|345, 126|35|4, 1346|25, 136|245, 136|25|4, 146|235, 16|2345, 16|235|4, 16|25|34.
a(7) = 31: 1234567, 123457|6, 12347|56, 12357|46, 1237|456, 1237|46|5, 12457|36, 1247|356, 1247|36|5, 1257|346, 127|3456, 127|346|5, 127|36|45, 13457|26, 1347|256, 1347|26|5, 1357|246, 137|2456, 137|246|5, 137|26|45, 1457|236, 147|2356, 147|236|5, 157|2346, 17|23456, 17|2346|5, 17|236|45, 147|26|35, 17|246|35, 17|26|345, 17|26|35|4.

Maple

b:= proc(n, m) option remember;
     `if`(m<n, b(n-1, m)*m+b(n-1, m+1), 1)
    end:
a:= n-> b(n, 0):
seq(a(n), n=0..31);

Crossrefs

Cf. A000110, A008277, A048993, A171367, A320964.

Sequence in context: A300660 A077212 A076836 * A296531 A389893 A220699

Adjacent sequences: A363068 A363069 A363070 * A363072 A363073 A363074

Keyword

nonn

Author

Alois P. Heinz, May 16 2023

Status

approved