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);
-
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).
A327712
Sum of multinomials M(n-k; p_1-1, ..., p_k-1), where p = (p_1, ..., p_k) ranges over all compositions of n into distinct parts (k is a composition length).
Original entry on oeis.org
1, 1, 1, 3, 3, 9, 29, 57, 135, 615, 2635, 6273, 25151, 82623, 525281, 2941047, 9100709, 38766777, 205155713, 902705793, 7714938567, 52987356783, 204844103977, 1042657233471, 5520661314689, 38159472253821, 211945677298567, 2404720648663335, 19773733727088813
Offset: 0
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with(combinat):
a:= n-> add(multinomial(n-nops(p), map(x-> x-1, p)[], 0), p=map(h->
permute(h)[], select(l-> nops(l)=nops({l[]}), partition(n)))):
seq(a(n), n=0..28);
# second Maple program:
a:= proc(m) option remember; local b; b:=
proc(n, i, j) option remember; `if`(i*(i+1)/2>=n,
`if`(n=0, (m-j)!*j!, b(n, i-1, j)+
b(n-i, min(n-i, i-1), j+1)/(i-1)!), 0)
end: b(m$2, 0):
end:
seq(a(n), n=0..28);
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a[m_] := a[m] = Module[{b}, b[n_, i_, j_] := b[n, i, j] = If[i(i + 1)/2 >= n, If[n == 0, (m - j)! j!, b[n, i - 1, j] + b[n - i, Min[n - i, i - 1], j + 1]/(i - 1)!], 0]; b[m, m, 0]];
a /@ Range[0, 28] (* Jean-François Alcover, May 10 2020, after 2nd Maple program *)
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 1-4 of 4 results.
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