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).
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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)*(i-1)!^j/j!
*combinat[multinomial](n, i$j, n-i*j), 0), j=0..n/i))
end:
a:= n-> b(n$2):
seq(a(n), n=0..25);
A372579
Number of permutations of [n] such that the number of cycles of length k is zero or equals k for every k.
Original entry on oeis.org
1, 1, 0, 0, 3, 15, 0, 0, 0, 2240, 22400, 0, 0, 4804800, 67267200, 0, 3405402000, 57891834000, 0, 0, 49497518070000, 1039447879470000, 0, 0, 0, 56947245360343962624, 1480628379368943028224, 0, 0, 4057662073660588368847872, 121729862209817651065436160, 0, 0, 0
Offset: 0
a(5) = 15 = 5*3: (1)(23)(45), (1)(24)(35), (1)(25)(34), ..., (1,2)(3,4)(5),
(1,3)(2,4)(5), (1,4)(2,3)(5).
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b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
add(`if`(j=0 or j=i, b(n-i*j, i-1)*(i-1)!^j/j!*
combinat[multinomial](n, i$j, n-i*j), 0), j=0..n/i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..35);
A374262
Number of permutations of [n] such that the number of cycles of length k is a multiple of k for every k.
Original entry on oeis.org
1, 1, 1, 1, 4, 16, 46, 106, 316, 3564, 27756, 141516, 556656, 6678816, 73015944, 521124696, 6144018336, 75200767776, 677927254176, 4642387894944, 75217104395136, 1167068528384256, 12348761954020416, 97377968145352896, 882819252604721664, 66882151986021043200
Offset: 0
a(4) = 4: (1)(2)(3)(4), (1,2)(3,4), (1,3)(2,4), (1,4)(2,3).
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b:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
add(combinat[multinomial](n, i$i*j, n-i^2*j)*
b(n-i^2*j, i-1)*(i-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);
Showing 1-4 of 4 results.