A286071
Number of permutations of [n] with nonincreasing cycle sizes.
Original entry on oeis.org
1, 1, 2, 5, 19, 85, 496, 3229, 25117, 215225, 2100430, 22187281, 261228199, 3284651245, 45163266604, 659277401525, 10380194835601, 172251467909809, 3057368096689690, 56867779157145769, 1122474190194034555, 23137433884903034501, 502874858021076645784
Offset: 0
a(3) = 5: (123), (132), (12)(3), (13)(2), (1)(2)(3).
a(4) = 19: (1234), (1243), (1324), (1342), (1423), (1432), (123)(4), (132)(4), (124)(3), (142)(3), (12)(34), (12)(3)(4), (134)(2), (143)(2), (13)(24), (13)(2)(4), (14)(23), (14)(2)(3), (1)(2)(3)(4).
-
b:= proc(n, i) option remember; `if`(n=0, 1, add((j-1)!*
b(n-j, j)*binomial(n-1, j-1), j=1..min(n, i)))
end:
a:= n-> b(n$2):
seq(a(n), n=0..30);
-
b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[(j - 1)!*b[n - j, j]*Binomial[n - 1, j - 1], {j, 1, Min[n, i]}]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 24 2018, translated from Maple *)
A286072
Number of permutations of [n] where adjacent cycles differ in size.
Original entry on oeis.org
1, 1, 1, 5, 16, 84, 512, 3572, 28080, 256820, 2553728, 28064776, 337319944, 4385615904, 61255920936, 921584068648, 14720437293952, 250190161426720, 4507229152534944, 85630125536152160, 1711040906290680448, 35969941361999955392, 790961860293623563648
Offset: 0
-
b:= proc(n, i) option remember; `if`(n=0, 1, add(`if`(i=j, 0,
(j-1)!*b(n-j, `if`(j>n-j, 0, j))*binomial(n-1, j-1)), j=1..n))
end:
a:= n-> b(n, 0):
seq(a(n), n=0..30);
-
b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[If[i == j, 0, (j - 1)!*b[n - j, If[j > n - j, 0, j]]*Binomial[n - 1, j - 1]], {j, 1, n}]];
a[n_] := b[n, 0];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 24 2018, translated from Maple *)
A286073
Number of permutations of [n] with decreasing cycle sizes.
Original entry on oeis.org
1, 1, 1, 4, 12, 60, 340, 2280, 17220, 151872, 1459584, 15624000, 182318400, 2316837600, 31596570720, 465582237120, 7283287851840, 121620647715840, 2149774858183680, 40196871701360640, 790002144844738560, 16364478334463078400, 354458730544573132800
Offset: 0
-
b:= proc(n, i) option remember;
`if`(n>i*(i+1)/2, 0, `if`(n=0, 1, b(n, i-1)+
`if`(i>n, 0, b(n-i, i-1)*(i-1)!*binomial(n-1, i-1))))
end:
a:= n-> b(n$2):
seq(a(n), n=0..30);
-
b[n_, i_] := b[n, i] = If[n > i*(i + 1)/2, 0, If[n == 0, 1, b[n, i - 1] + If[i > n, 0, b[n - i, i - 1]*(i - 1)!*Binomial[n - 1, i - 1]]]];
a[n_] := b[n, n];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 24 2018, translated from Maple *)
A286074
Number of permutations of [n] with nondecreasing cycle sizes.
Original entry on oeis.org
1, 1, 2, 4, 13, 45, 250, 1342, 10085, 76165, 715588, 6786108, 78636601, 896672473, 12112535378, 163963519810, 2534311844905, 39211836764457, 688584972407680, 12003902219337760, 234324625117308533, 4571805253649981173, 98183221386947058286
Offset: 0
-
b:= proc(n, i) option remember; `if`(n=0, 1, add(
(j-1)!*b(n-j, j)*binomial(n-1, j-1), j=i..n))
end:
a:= n-> b(n, 1):
seq(a(n), n=0..30);
-
b[n_, i_] := b[n, i] = If[n == 0, 1, Sum[(j - 1)!*b[n - j, j]*Binomial[n - 1, j - 1], {j, i, n}]];
a[n_] := b[n, 1];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 25 2018, translated from Maple *)
A286075
Number of permutations of [n] with increasing cycle sizes.
Original entry on oeis.org
1, 1, 1, 3, 8, 38, 182, 1194, 7932, 69192, 591936, 6286272, 66914880, 840036960, 10567285920, 154755036000, 2246755924800, 37283584936320, 618705247829760, 11472473012232960, 212762383625594880, 4386435706887413760, 89954629722500659200, 2030764767987849062400
Offset: 0
-
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i>n, 0,
b(n, i+1)+b(n-i, i+1)*(i-1)!*binomial(n-1, i-1)))
end:
a:= n-> b(n, 1):
seq(a(n), n=0..30);
-
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i > n, 0, b[n, i + 1] + b[n - i, i + 1]*(i - 1)!*Binomial[n - 1, i - 1]]];
a[n_] := b[n, 1];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 28 2018, from Maple *)
A286077
Number of permutations of [n] with a strongly unimodal cycle size list.
Original entry on oeis.org
1, 1, 1, 5, 16, 80, 468, 3220, 24436, 218032, 2114244, 22759788, 267150264, 3413938512, 46668380592, 690881123856, 10841100147072, 181434400544160, 3215124610986240, 60280035304993920, 1186176116251848960, 24624604679704053120, 534223121657911528320
Offset: 0
-
b:= proc(n, i, t) option remember; `if`(t=0 and n>i*(i-1)/2, 0,
`if`(n=0, 1, add(b(n-j, j, 0)*binomial(n-1, j-1)*
(j-1)!, j=1..min(n, i-1))+`if`(t=1, add(b(n-j, j, 1)*
binomial(n-1, j-1)*(j-1)!, j=i+1..n), 0)))
end:
a:= n-> b(n, 0, 1):
seq(a(n), n=0..30);
-
b[n_, i_, t_] := b[n, i, t] = If[t == 0 && n > i*(i-1)/2, 0, If[n == 0, 1, Sum[b[n-j, j, 0]*Binomial[n-1, j-1]*(j-1)!, {j, 1, Min[n, i-1]}] + If[t == 1, Sum[b[n-j, j, 1]*Binomial[n-1, j-1]*(j-1)!, {j, i+1, n}], 0]]];
a[n_] := b[n, 0, 1];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, May 28 2018, from Maple *)
A275679
Number of set partitions of [n] with alternating block size parities.
Original entry on oeis.org
1, 1, 1, 4, 3, 20, 43, 136, 711, 1606, 12653, 36852, 250673, 1212498, 6016715, 45081688, 196537387, 1789229594, 8963510621, 76863454428, 512264745473, 3744799424978, 32870550965259, 219159966518160, 2257073412153459, 15778075163815474, 165231652982941085
Offset: 0
a(3) = 4: 123, 12|3, 13|2, 1|23.
a(4) = 3: 1234, 1|23|4, 1|24|3.
a(5) = 20: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 12|3|45, 1345|2, 134|25, 135|24, 13|245, 13|2|45, 145|23, 14|235, 15|234, 1|2345, 14|2|35, 15|2|34.
Cf.
A003724,
A005046,
A007837,
A038041,
A275309,
A275310,
A275311,
A275312,
A275313,
A286076,
A361804.
-
b:= proc(n, t) option remember; `if`(n=0, 1, add(
`if`((i+t)::odd, b(n-i, 1-t)*binomial(n-1, i-1), 0), i=1..n))
end:
a:= n-> `if`(n=0, 1, b(n, 0)+b(n, 1)):
seq(a(n), n=0..35);
-
b[n_, t_] := b[n, t] = If[n==0, 1, Sum[If[OddQ[i+t], b[n-i, 1-t] * Binomial[n-1, i-1], 0], {i, 1, n}]]; a[n_] := If[n==0, 1, b[n, 0] + b[n, 1]]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Feb 27 2017, translated from Maple *)
Showing 1-7 of 7 results.
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