A361084 Number of partitions of [n] such that in each block the smallest element and the largest element have opposite parities.
1, 0, 1, 0, 3, 2, 17, 28, 171, 430, 2617, 8496, 54739, 214714, 1477153, 6743204, 49550011, 256645926, 2010328585, 11602635128, 96590823907, 612918061426, 5404902119025, 37319203169580, 347468152001739, 2589081441826334, 25375080898848729, 202668739104752960
Offset: 0
Keywords
Examples
a(0) = 1: () the empty partition. a(1) = 0. a(2) = 1: 12. a(3) = 0. a(4) = 3: 1234, 12|34, 14|23. a(5) = 2: 134|25, 14|235. a(6) = 17: 123456, 1234|56, 1236|45, 124|356, 1256|34, 12|3456, 12|34|56, 12|36|45, 1346|25, 136|245, 1456|23, 146|235, 14|23|56, 16|2345, 16|23|45, 14|25|36, 16|25|34. a(7) = 28: 12356|47, 1236|457, 12|356|47, 12|36|457, 13456|27, 1346|257, 134|2567, 134|25|67, 134|27|56, 1356|247, 136|2457, 136|25|47, 136|27|45, 1456|237, 146|2357, 14|23567, 14|235|67, 14|237|56, 156|2347, 16|23457, 156|23|47, 16|235|47, 16|237|45, 16|23|457, 14|257|36, 14|27|356, 156|27|34, 16|257|34.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..165
- Wikipedia, Partition of a set
Programs
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Maple
b:= proc(n, x, y, u, v) option remember; `if`(x+v>n, 0, `if`(n=0, 1, `if`(y=0, 0, b(n-1, v, u, y-1, x+1)*y)+b(n-1, v, u, y, x+1)+ `if`(v=0, 0, b(n-1, v-1, u+1, y, x)*v)+b(n-1, v, u, y, x)*(u+x))) end: a:= n-> b(n, 0$4): seq(a(n), n=0..30);
Formula
a(n) mod 2 = A059841(n).