A364406
Number of permutations of [n] such that the minimal element of each cycle is also its length.
Original entry on oeis.org
1, 1, 0, 1, 0, 0, 6, 6, 0, 0, 720, 2160, 9360, 19440, 30240, 3659040, 21772800, 228614400, 1632960000, 11125900800, 73025971200, 1708337433600, 15442053580800, 254260755302400, 3318429200486400, 46929444097536000, 546974781889536000, 7312714579602432000
Offset: 0
a(0) = 1: () the empty permutation.
a(1) = 1: (1).
a(3) = 1: (1)(23).
a(6) = 6: (1)(24)(356), (1)(24)(365), (1)(25)(346), (1)(25)(364),
(1)(26)(345), (1)(26)(354).
a(7) = 6: (1)(23)(4567), (1)(23)(4576), (1)(23)(4657), (1)(23)(4675),
(1)(23)(4756), (1)(23)(4765).
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b:= proc(n, i) option remember; `if`(i*(i+1)/2n+1, 0, b(n-i, i-1)*binomial(n-i, i-1)*(i-1)!)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..33);
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b[n_, i_] := b[n, i] = If[i*(i + 1)/2 < n, 0, If[n == 0, 1, b[n, i - 1] + If[2*i > n + 1, 0, b[n - i, i - 1]*Binomial[n - i, i - 1]*(i - 1)!]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Dec 05 2023, after Alois P. Heinz *)
A364277
Number of permutations of [n] such that no cycle contains its length as an element.
Original entry on oeis.org
1, 0, 0, 1, 4, 24, 138, 1032, 8160, 75600, 751680, 8436960, 100679040, 1327052160, 18525024000, 280451808000, 4477627123200, 76690072166400, 1377634946688000, 26328977260185600, 525869478021888000, 11092929741653760000, 243781091314016256000, 5628622656645660672000
Offset: 0
a(3) = 1: (13)(2).
a(4) = 4: (124)(3), (142)(3), (13)(2)(4), (14)(2)(3).
A364281
Number of permutations of [n] with distinct cycle lengths such that each cycle contains exactly one cycle length as an element.
Original entry on oeis.org
1, 1, 1, 4, 10, 48, 252, 1584, 10800, 93600, 823680, 8588160, 93381120, 1158312960, 14805504000, 215028172800, 3159494553600, 51973589606400, 873152856576000, 16058241239040000, 300754643245056000, 6159522883497984000, 127439374149255168000
Offset: 0
a(3) = 4: (123), (132), (13)(2), (1)(23).
a(4) = 10: (1234), (1243), (1324), (1342), (1423), (1432), (124)(3),
(142)(3), (1)(234), (1)(243).
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a:= proc(m) option remember; local b; b:=
proc(n, i, p) option remember; `if`(i*(i+1)/2
-
a[m_] := a[m] = Module[{b}, b[n_, i_, p_] := b[n, i, p] = If[i(i+1)/2 < n, 0, If[n == 0, p!*(m - p)!, b[n, i - 1, p] + b[n - i, Min[n - i, i - 1], p - 1]]]; b[m, m, m]];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Oct 21 2023, after Alois P. Heinz *)
A364278
Number of partitions of [n] with distinct block sizes such that no block contains its size as an element.
Original entry on oeis.org
1, 0, 0, 1, 1, 4, 24, 47, 153, 669, 5628, 13554, 61747, 247170, 1539565, 16979571, 53166394, 268393296, 1382097160, 7831424654, 59720804940, 917256305956, 3326800474687, 20441030261195, 112690616749302, 773175024537549, 5164903931159843, 52976603588044961
Offset: 0
a(0) = 1: () the empty partition.
a(3) = 1: 13|2.
a(4) = 1: 124|3.
a(5) = 4: 1235|4, 124|35, 125|34, 13|245.
A364283
Number of permutations of [n] with distinct cycle lengths such that each cycle contains exactly one cycle length different from its own as an element.
Original entry on oeis.org
1, 0, 0, 1, 2, 12, 60, 408, 2640, 24480, 208080, 2262960, 23950080, 307359360, 3835641600, 57400358400, 825160089600, 13909727462400, 229664981145600, 4310966499840000, 79428141112320000, 1658163790483200000, 33795850208440320000, 770528520983789568000
Offset: 0
a(3) = 1: (13)(2).
a(4) = 2: (124)(3), (142)(3).
a(5) = 12: (1235)(4), (1253)(4), (1325)(4), (1352)(4), (1523)(4), (1532)(4),
(124)(35), (142)(35), (125)(34), (152)(34), (13)(245), (13)(254).
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f:= proc(n) option remember; `if`(n<2, 1-n, (n-1)*(f(n-1)+f(n-2))) end:
a:= proc(m) option remember; local b; b:=
proc(n, i, p) option remember; `if`(i*(i+1)/2
Showing 1-5 of 5 results.