A374262
Number of permutations of [n] such that the number of cycles of length k is a multiple of k for every k.
Original entry on oeis.org
1, 1, 1, 1, 4, 16, 46, 106, 316, 3564, 27756, 141516, 556656, 6678816, 73015944, 521124696, 6144018336, 75200767776, 677927254176, 4642387894944, 75217104395136, 1167068528384256, 12348761954020416, 97377968145352896, 882819252604721664, 66882151986021043200
Offset: 0
a(4) = 4: (1)(2)(3)(4), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3).
-
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-1)!^(i*j)/(i*j)!, j=0..n/i^2))
end:
a:= n-> b(n$2):
seq(a(n), n=0..25);
A374319
Number of partitions of [n] such that the number of blocks of size k is zero or a divisor of k for every k.
Original entry on oeis.org
1, 1, 1, 4, 8, 31, 82, 274, 1626, 5135, 26751, 125489, 1020692, 4333707, 31083613, 132960104, 1323145731, 8282668312, 70017330978, 423293287673, 3135764479898, 30762429056580, 269133472001923, 2185746568531948, 15121514389566421, 147045774699171957
Offset: 0
a(0) = 1: the empty partition.
a(1) = 1: 1.
a(2) = 1: 12.
a(3) = 4: 123, 12|3, 13|2, 1|23.
a(4) = 8: 1234, 123|4, 124|3, 12|34, 134|2, 13|24, 14|23, 1|234.
a(5) = 31: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 12|34|5, 12|35|4, 12|3|45, 1345|2, 134|25, 135|24, 13|245, 13|24|5, 13|25|4, 13|2|45, 145|23, 14|235, 14|23|5, 15|234, 1|2345, 15|23|4, 1|23|45, 14|25|3, 14|2|35, 15|24|3, 1|24|35, 15|2|34, 1|25|34.
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(`if`(j=0 or irem(i, j)=0, b(n-i*j, i-1)/j!*
combinat[multinomial](n, i$j, n-i*j), 0), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..27);
A374321
Number of partitions of [n] such that the number of blocks of size k is zero or equals k for every k.
Original entry on oeis.org
1, 1, 0, 0, 3, 15, 0, 0, 0, 280, 2800, 0, 0, 600600, 8408400, 0, 2627625, 44669625, 0, 0, 38192529375, 802043116875, 0, 0, 0, 1508282884484376, 39215354996593776, 0, 0, 107469680368165243128, 3224090411044957293840, 0, 0, 0, 76290792475347121351680
Offset: 0
a(0) = 1: the empty partition.
a(1) = 1: 1.
a(4) = 3: 12|34, 13|24, 14|23.
a(5) = 15: 12|34|5, 12|35|4, 12|3|45, 13|24|5, 13|25|4, 13|2|45, 14|23|5, 15|23|4, 1|23|45, 14|25|3, 14|2|35, 15|24|3, 1|24|35, 15|2|34, 1|25|34.
a(9) = 280: 123|456|789, 123|457|689, 123|458|679, 123|459|678, ..., 169|278|345, 178|269|345, 179|268|345, 189|267|345.
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b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i<1, 0, add(`if`(j=0 or j=i, b(n-i*j, i-1)/j!*
combinat[multinomial](n, i$j, n-i*j), 0), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..35);
A374329
Number of partitions of [n] such that the number of blocks of size k is a multiple or a divisor of k for every k.
Original entry on oeis.org
1, 1, 2, 5, 15, 52, 178, 702, 3160, 15267, 72904, 401139, 2473320, 15932022, 99025826, 641593206, 4570555035, 35589342752, 281312386194, 2198910615580, 17652574088975, 154057648160882, 1437155645037380, 13320654235409156, 121511456568711443, 1135554227462348853
Offset: 0
a(6) = 178 = 203 - 25 counts all partitions of [6] with the exception of 123|456, 124|356, 125|346, 126|345, 12|34|56, 12|35|46, 12|36|45, 134|256, 135|246, 136|245, 13|24|56, 13|25|46, 13|26|45, 145|236, 146|235, 14|23|56, 156|234, 15|23|46, 16|23|45, 14|25|36, 14|26|35, 15|24|36, 16|24|35, 15|26|34, 16|25|34.
-
b:= proc(n, i) option remember; `if`(n=0 or i=1, 1, add(
`if`(irem(j, i)=0 or irem(i, j)=0, b(n-i*j, i-1)/j!*
combinat[multinomial](n, i$j, n-i*j), 0), j=0..n/i))
end:
a:= n-> b(n$2):
seq(a(n), n=0..25);
Showing 1-4 of 4 results.