A372545
Number of permutations of [n] such that the number of cycles of length k is a multiple or a divisor of k for every k.
Original entry on oeis.org
1, 1, 2, 6, 24, 120, 665, 4655, 37660, 345660, 3373629, 37109919, 443171498, 5761229474, 79709485141, 1199252731963, 19237203662248, 327101074802216, 5848216651372953, 111064609625430747, 2222478622302320382, 46709011248199791062, 1022898268873467547769
Offset: 0
a(6) = 665 = 720 - 55 counts all permutations of [6] with the exception of 15 permutations of type (12)(34)(56) and 40 permutations of type (123)(456).
-
b:= proc(n, i) option remember; `if`(n=0 or i=1, 1, add(`if`(
irem(j, i)=0 or irem(i, j)=0, b(n-i*j, i-1)*(i-1)!^j/j!
*combinat[multinomial](n, i$j, n-i*j), 0), j=0..n/i))
end:
a:= n-> b(n$2):
seq(a(n), n=0..25);
A374319
Number of partitions of [n] such that the number of blocks of size k is zero or a divisor of k for every k.
Original entry on oeis.org
1, 1, 1, 4, 8, 31, 82, 274, 1626, 5135, 26751, 125489, 1020692, 4333707, 31083613, 132960104, 1323145731, 8282668312, 70017330978, 423293287673, 3135764479898, 30762429056580, 269133472001923, 2185746568531948, 15121514389566421, 147045774699171957
Offset: 0
a(0) = 1: the empty partition.
a(1) = 1: 1.
a(2) = 1: 12.
a(3) = 4: 123, 12|3, 13|2, 1|23.
a(4) = 8: 1234, 123|4, 124|3, 12|34, 134|2, 13|24, 14|23, 1|234.
a(5) = 31: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 12|34|5, 12|35|4, 12|3|45, 1345|2, 134|25, 135|24, 13|245, 13|24|5, 13|25|4, 13|2|45, 145|23, 14|235, 14|23|5, 15|234, 1|2345, 15|23|4, 1|23|45, 14|25|3, 14|2|35, 15|24|3, 1|24|35, 15|2|34, 1|25|34.
-
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(`if`(j=0 or irem(i, j)=0, b(n-i*j, i-1)/j!*
combinat[multinomial](n, i$j, n-i*j), 0), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..27);
A374320
Number of partitions of [n] such that the number of blocks of size k is a multiple of k for every k.
Original entry on oeis.org
1, 1, 1, 1, 4, 16, 46, 106, 316, 1604, 8156, 33716, 125456, 1073216, 10233224, 69873896, 364469561, 2296961801, 19124734801, 147200743489, 960313414036, 6422446261456, 52845891370966, 461844834503746, 3779922654292324, 31131912140021452, 296987899271509252
Offset: 0
a(0) = 1: the empty partition.
a(1) = 1: 1.
a(2) = 1: 1|2.
a(3) = 1: 1|2|3.
a(4) = 4: 12|34, 13|24, 14|23, 1|2|3|4.
a(5) = 16: 12|34|5, 12|35|4, 12|3|45, 13|24|5, 13|25|4, 13|2|45, 14|23|5, 15|23|4, 1|23|45, 14|25|3, 14|2|35, 15|24|3, 1|24|35, 15|2|34, 1|25|34, 1|2|3|4|5.
a(9) = 1604: 123|456|789, 123|457|689, 123|458|679, 123|459|678, ..., 1|2|3|49|5|6|78, 1|2|3|4|59|6|78, 1|2|3|4|5|69|78, 1|2|3|4|5|6|7|8|9.
-
b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
add(combinat[multinomial](n, i$i*j, n-i^2*j)*
b(n-i^2*j, i-1)/(i*j)!, j=0..n/i^2))
end:
a:= n-> b(n$2):
seq(a(n), n=0..28);
A374321
Number of partitions of [n] such that the number of blocks of size k is zero or equals k for every k.
Original entry on oeis.org
1, 1, 0, 0, 3, 15, 0, 0, 0, 280, 2800, 0, 0, 600600, 8408400, 0, 2627625, 44669625, 0, 0, 38192529375, 802043116875, 0, 0, 0, 1508282884484376, 39215354996593776, 0, 0, 107469680368165243128, 3224090411044957293840, 0, 0, 0, 76290792475347121351680
Offset: 0
a(0) = 1: the empty partition.
a(1) = 1: 1.
a(4) = 3: 12|34, 13|24, 14|23.
a(5) = 15: 12|34|5, 12|35|4, 12|3|45, 13|24|5, 13|25|4, 13|2|45, 14|23|5, 15|23|4, 1|23|45, 14|25|3, 14|2|35, 15|24|3, 1|24|35, 15|2|34, 1|25|34.
a(9) = 280: 123|456|789, 123|457|689, 123|458|679, 123|459|678, ..., 169|278|345, 178|269|345, 179|268|345, 189|267|345.
-
b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i<1, 0, add(`if`(j=0 or j=i, b(n-i*j, i-1)/j!*
combinat[multinomial](n, i$j, n-i*j), 0), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..35);
Showing 1-4 of 4 results.