A362362
Number of permutations of [n] such that each cycle contains its length as an element.
Original entry on oeis.org
1, 1, 1, 3, 8, 36, 174, 1104, 7440, 62640, 545040, 5649840, 60681600, 748621440, 9518342400, 136758585600, 2009451628800, 32848492723200, 549241915622400, 10066913176320000, 188293339922688000, 3832031198451456000, 79291640831090688000, 1771146970953744384000
Offset: 0
a(3) = 3: (123), (132), (1)(23).
a(4) = 8: (1234), (1243), (1324), (1342), (1423), (1432), (1)(234), (1)(243).
-
a:= n-> add((n-nops(p))!, p=select(l-> nops(l)=
nops({l[]}), combinat[partition](n))):
seq(a(n), n=0..24);
# second Maple program:
b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 b(n$3):
seq(a(n), n=0..24);
-
b[n_, i_, p_] := b[n, i, p] = If[i*(i + 1)/2 < n, 0, If[n == 0, p!, b[n, i - 1, p] + b[n - i, Min[n - i, i - 1], p - 1]]];
a[n_] := b[n, n, n];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Nov 15 2023, from second Maple program *)
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).
-
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);
-
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 *)
A364279
Number of permutations of [n] with distinct cycle lengths such that no cycle contains its length as an element.
Original entry on oeis.org
1, 0, 0, 1, 2, 12, 86, 546, 4284, 39588, 416988, 4378848, 54297504, 695592000, 9840307680, 149031686880, 2387863575360, 40338090711360, 736126007279040, 13938942123429120, 279358800902737920, 5894877845100625920, 129943826126987765760, 2985640822908446976000
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).
A363881
Number of partitions of [n] such that no block contains its size as an element.
Original entry on oeis.org
1, 0, 0, 1, 3, 13, 50, 230, 1110, 5787, 32335, 191950, 1206247, 7997702, 55733468, 406952888, 3105706421, 24710573792, 204547052598, 1758110677909, 15663244043627, 144412976181189, 1375896762258868, 13528184875421816, 137098357090429007, 1430440534060723253
Offset: 0
a(0) = 1: () the empty partition.
a(3) = 1: 13|2.
a(4) = 3: 124|3, 13|2|4, 14|2|3.
a(5) = 13: 1235|4, 124|35, 124|3|5, 125|34, 125|3|4, 13|245, 13|2|45, 13|2|4|5, 145|2|3, 14|2|35, 14|2|3|5, 15|2|34, 15|2|3|4.
a(6) = 50: 12346|5, 1235|46, 1235|4|6, 1236|45, ..., 15|2|3|4|6, 16|2|35|4, 16|2|3|45, 16|2|3|4|5.
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.
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