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 *)
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).
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).
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).
-
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 *)
A364207
Number of partitions of [n] such that the minimal element of each block is also its size.
Original entry on oeis.org
1, 1, 0, 1, 0, 0, 3, 1, 0, 0, 60, 45, 53, 24, 7, 12601, 15120, 33390, 55710, 66522, 86037, 37907754, 63130067, 202203684, 511378789, 1421634137, 2566309603, 5855352202, 2064277450957, 4418631559288, 18485494082571, 61020702809287, 232959438927000, 783244248553960
Offset: 0
a(0) = 1: () the empty partition.
a(1) = 1: 1.
a(3) = 1: 1|23.
a(6) = 3: 1|24|356, 1|25|346, 1|26|345.
a(7) = 1: 1|23|4567.
a(10) = 60: 1|25|367|489(10), 1|25|368|479(10), 1|25|369|478(10), ..., 1|28|39(10)|4567, 1|29|38(10)|4567, 1|2(10)|389|4567.
a(14) = 7: 1|23|4568|79(10)(11)(12)(13)(14), 1|23|4569|78(10)(11)(12)(13)(14), 1|23|456(10)|789(11)(12)(13)(14), 1|23|456(11)|789(10)(12)(13)(14), 1|23|456(12)|789(10)(11)(13)(14), 1|23|456(13)|789(10)(11)(12)(14), 1|23|456(14)|789(10)(11)(12)(13).
-
b:= proc(n, i) option remember; `if`(i*(i+1)/2n or i>n-i+1, 0, b(n-i, i-1)*binomial(n-i, i-1))))
end:
a:= n-> b(n$2):
seq(a(n), n=0..33); # Alois P. Heinz, Jul 22 2023
-
b[n_, i_] := b[n, i] = If[i(i+1)/2 < n, 0, If[n == 0, 1, b[n, i-1] + If[i > n || i > n-i+1, 0, b[n-i, i-1]*Binomial[n-i, i-1]]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 33}] (* Jean-François Alcover, Oct 20 2023, after Alois P. Heinz *)
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).
-
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 1-6 of 6 results.
Comments