A374292
Number of permutations of [n] such that the number of cycles of length k is zero or a divisor of k for every k.
Original entry on oeis.org
1, 1, 1, 5, 17, 89, 474, 3324, 28440, 253448, 2476700, 26876420, 328110540, 4207321260, 58468831680, 877439227560, 14214209548560, 239870470655760, 4285924637475600, 81381169697904720, 1636049164466934000, 34301061146870607600, 750389221227585139200
Offset: 0
a(3) = 5: (1)(2,3), (1,2)(3), (1,3)(2), (1,2,3), (1,3,2).
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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)*(i-1)!^j/j!*
combinat[multinomial](n, i$j, n-i*j), 0), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..25);
A374320
Number of partitions of [n] such that the number of blocks of size k is a multiple of k for every k.
Original entry on oeis.org
1, 1, 1, 1, 4, 16, 46, 106, 316, 1604, 8156, 33716, 125456, 1073216, 10233224, 69873896, 364469561, 2296961801, 19124734801, 147200743489, 960313414036, 6422446261456, 52845891370966, 461844834503746, 3779922654292324, 31131912140021452, 296987899271509252
Offset: 0
a(0) = 1: the empty partition.
a(1) = 1: 1.
a(2) = 1: 1|2.
a(3) = 1: 1|2|3.
a(4) = 4: 12|34, 13|24, 14|23, 1|2|3|4.
a(5) = 16: 12|34|5, 12|35|4, 12|3|45, 13|24|5, 13|25|4, 13|2|45, 14|23|5, 15|23|4, 1|23|45, 14|25|3, 14|2|35, 15|24|3, 1|24|35, 15|2|34, 1|25|34, 1|2|3|4|5.
a(9) = 1604: 123|456|789, 123|457|689, 123|458|679, 123|459|678, ..., 1|2|3|49|5|6|78, 1|2|3|4|59|6|78, 1|2|3|4|5|69|78, 1|2|3|4|5|6|7|8|9.
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b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
add(combinat[multinomial](n, i$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$2):
seq(a(n), n=0..28);
A374321
Number of partitions of [n] such that the number of blocks of size k is zero or equals k for every k.
Original entry on oeis.org
1, 1, 0, 0, 3, 15, 0, 0, 0, 280, 2800, 0, 0, 600600, 8408400, 0, 2627625, 44669625, 0, 0, 38192529375, 802043116875, 0, 0, 0, 1508282884484376, 39215354996593776, 0, 0, 107469680368165243128, 3224090411044957293840, 0, 0, 0, 76290792475347121351680
Offset: 0
a(0) = 1: the empty partition.
a(1) = 1: 1.
a(4) = 3: 12|34, 13|24, 14|23.
a(5) = 15: 12|34|5, 12|35|4, 12|3|45, 13|24|5, 13|25|4, 13|2|45, 14|23|5, 15|23|4, 1|23|45, 14|25|3, 14|2|35, 15|24|3, 1|24|35, 15|2|34, 1|25|34.
a(9) = 280: 123|456|789, 123|457|689, 123|458|679, 123|459|678, ..., 169|278|345, 178|269|345, 179|268|345, 189|267|345.
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b:= proc(n, i) option remember; `if`(n=0, 1,
`if`(i<1, 0, add(`if`(j=0 or j=i, b(n-i*j, i-1)/j!*
combinat[multinomial](n, i$j, n-i*j), 0), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..35);
A374329
Number of partitions of [n] such that the number of blocks of size k is a multiple or a divisor of k for every k.
Original entry on oeis.org
1, 1, 2, 5, 15, 52, 178, 702, 3160, 15267, 72904, 401139, 2473320, 15932022, 99025826, 641593206, 4570555035, 35589342752, 281312386194, 2198910615580, 17652574088975, 154057648160882, 1437155645037380, 13320654235409156, 121511456568711443, 1135554227462348853
Offset: 0
a(6) = 178 = 203 - 25 counts all partitions of [6] with the exception of 123|456, 124|356, 125|346, 126|345, 12|34|56, 12|35|46, 12|36|45, 134|256, 135|246, 136|245, 13|24|56, 13|25|46, 13|26|45, 145|236, 146|235, 14|23|56, 156|234, 15|23|46, 16|23|45, 14|25|36, 14|26|35, 15|24|36, 16|24|35, 15|26|34, 16|25|34.
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b:= proc(n, i) option remember; `if`(n=0 or i=1, 1, add(
`if`(irem(j, i)=0 or irem(i, j)=0, b(n-i*j, i-1)/j!*
combinat[multinomial](n, i$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.