A362362 -id:A362362 - cp's OEIS frontend

A326493 Sum of multinomials M(n-k; p_1-1, ..., p_k-1), where p = (p_1, ..., p_k) ranges over all partitions of n into distinct parts (k is a partition length).

1, 1, 1, 2, 2, 5, 9, 21, 38, 146, 322, 902, 3106, 8406, 35865, 123321, 393691, 1442688, 7310744, 23471306, 129918661, 500183094, 2400722981, 9592382321, 47764284769, 280267554944, 1247781159201, 7620923955225, 36278364107926, 189688942325418, 1124492015730891

Offset: 0

Alois P. Heinz, Sep 22 2019

nonn

Number of partitions of [n] such that each block contains its size as an element. So the block sizes have to be distinct. a(6) = 9: 123456, 12|3456, 1345|26, 1346|25, 1456|23, 1|23456, 1|24|356, 1|25|346, 1|26|345.

Alois P. Heinz, Table of n, a(n) for n = 0..731
Wikipedia, Multinomial coefficients
Wikipedia, Partition (number theory)
Wikipedia, Partition of a set

Cf. A000110, A007837, A327711, A327712, A362362, A363881, A364207, A364278.

with(combinat):
a:= n-> add(multinomial(n-nops(p), map(x-> x-1, p)[], 0),
        p=select(l-> nops(l)=nops({l[]}), partition(n))):
seq(a(n), n=0..30);
# second Maple program:
b:= proc(n, i, p) option remember; `if`(i*(i+1)/2 b(n$3):
seq(a(n), n=0..31);

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]/(i-1)!]];
a[n_] := b[n, n, n];
a /@ Range[0, 31] (* Jean-François Alcover, Dec 09 2020, after Alois P. Heinz *)

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

Alois P. Heinz, Jul 22 2023

nonn

			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).

Alois P. Heinz, Table of n, a(n) for n = 0..474
Wikipedia, Permutation

Cf. A000142, A007837, A188431, A362362, A364207, A364277 A364279 A364281 A364283.

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 *)

A179973 Number of permutations of [n] whose cycle lengths are nondecreasing when cycles are ordered by their minima and these minima are {1..k} (for some k <= n).

Original entry on oeis.org

1, 1, 2, 4, 12, 42, 216, 1200, 8664, 66384, 612264, 5910024, 66723384, 776642664, 10311400344, 141065450904, 2153769250584, 33743736435864, 583781959921944, 10308436641381144, 198863818304824344, 3914117125411211544, 83301822014343774744, 1805447764831655109144

Offset: 0

Alford Arnold, Aug 05 2010

nonn

The original name was: Row sums of A179972 and also of A179974.

			a(4) = 12 = 6 + 2 + 2 + 1 + 1: (1234), (1243), (1324), (1342), (1423), (1432),
  (13)(24), (14)(23), (1)(234), (1)(243), (1)(2)(34), (1)(2)(3)(4).

Alois P. Heinz, Table of n, a(n) for n = 0..450

Cf. A000041, A000142, A000522, A004086, A053529, A179972, A179974, A327711, A362362, A364128.

a:= n-> add((n-nops(p))!, p=combinat[partition](n)):
seq(a(n), n=0..24);  # Alois P. Heinz, Jul 09 2023
# second Maple program:
b:= proc(n, i, p) option remember; `if`(n=0 or i=1,
     (p-n)!, b(n, i-1, p)+b(n-i, min(n-i, i), p-1))
    end:
a:= n-> b(n$3):
seq(a(n), n=0..24);  # Alois P. Heinz, Jul 09 2023

b[n_, i_, p_] := b[n, i, p] = If[n == 0 || i == 1, (p - n)!, b[n, i - 1, p] + b[n - i, Min[n - i, i], p - 1]];
a[n_] := b[n, n, n];
Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Aug 16 2023, after Alois P. Heinz *)

From Alois P. Heinz, Jul 09 2023: (Start)
a(n) = Sum_{lambda in partitions(n)} (n - |lambda|)!.
Limit_{n->oo} A004086(a(n))/10^A055642(a(n)) = A364128. (End)

Edited by R. J. Mathar, May 17 2016

a(0), a(9)-a(23) and new name from Alois P. Heinz, Jul 09 2023

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

Alois P. Heinz, Jul 17 2023

nonn

			a(3) = 1: (13)(2).
a(4) = 4: (124)(3), (142)(3), (13)(2)(4), (14)(2)(3).

Wikipedia, Permutation

Cf. A362362, A363881, A364279, A364406.

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

Alois P. Heinz, Jul 17 2023

nonn

			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).

Wikipedia, Permutation

Cf. A000009, A007838, A362362, A364277, A364278, A364281, A364406.

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

Alois P. Heinz, Jul 17 2023

nonn

			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).

Alois P. Heinz, Table of n, a(n) for n = 0..450
Wikipedia, Permutation

Cf. A000009, A000166, A007838, A362362, A364277, A364279, A364281, A364282, A364406.

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

cp's OEIS Frontend

A326493 Sum of multinomials M(n-k; p_1-1, ..., p_k-1), where p = (p_1, ..., p_k) ranges over all partitions of n into distinct parts (k is a partition length).

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A364406 Number of permutations of [n] such that the minimal element of each cycle is also its length.

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A179973 Number of permutations of [n] whose cycle lengths are nondecreasing when cycles are ordered by their minima and these minima are {1..k} (for some k <= n).

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A364277 Number of permutations of [n] such that no cycle contains its length as an element.

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A364279 Number of permutations of [n] with distinct cycle lengths such that no cycle contains its length as an element.

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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.

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Programs

Maple