A327711 -id:A327711 - 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 *)

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

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

Alois P. Heinz, Sep 22 2019

nonn

Number of partitions of [n] with distinct block sizes such that each block contains exactly one block size as an element. a(5) = 9: 12345, 1235|4, 124|35, 125|34, 12|345, 134|25, 135|24, 13|245, 1|2345.

Alois P. Heinz, Table of n, a(n) for n = 0..706
Wikipedia, Multinomial coefficients
Wikipedia, Partition (number theory)
Wikipedia, Partition of a set

Cf. A026898, A326493, A327711, A364281.

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);

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 *)

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

Alois P. Heinz, Jul 13 2023

nonn

The block sizes are distinct as a consequence of the definition.

There are A188431(n) different block size configurations for a given n. - John Tyler Rascoe, Jul 19 2023

			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).

Alois P. Heinz, Table of n, a(n) for n = 0..735
Wikipedia, Partition of a set

Cf. A000110, A007837, A188431, A326493, A327711, A364406.

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 *)

A327729 a(n) = Sum_{p} M(n-k; p_1-1, ..., p_k-1) * Product_{j=1..k} a(p_j), where p = (p_1, ..., p_k) ranges over all partitions of n into smaller parts (k is a partition length and M is a multinomial).

Original entry on oeis.org

1, 1, 2, 6, 18, 90, 414, 2892, 18342, 155124, 1265130, 13413240, 129656286, 1564538796, 18285385518, 255345207156, 3378398348214, 52931303772912, 797460543143154, 13926097774972152, 234050020177159926, 4466082284967035124, 83159771376289666806

Offset: 1

Alois P. Heinz, Sep 23 2019

nonn

The formula is a generalization of the formula given in A327643.

Alois P. Heinz, Table of n, a(n) for n = 1..460
Wikipedia, Multinomial coefficients
Wikipedia, Partition (number theory)

Cf. A327643, A327711.

with(combinat):
a:= proc(n) option remember; `if`(n<2, 1, add(mul(a(i), i=p)
      *multinomial(n-nops(p), map(x-> x-1, p)[]),
       p=select(x-> nops(x)>1, partition(n))))
    end:
seq(a(n), n=1..24);
# second Maple program:
b:= proc(n, p, i) option remember; `if`(n=0, p!, `if`(i<1, 0,
      b(n, p, i-1) +a(i)*b(n-i, p-1, min(n-i, i))/(i-1)!))
    end:
a:= n-> `if`(n<2, 1, b(n$2, n-1)):
seq(a(n), n=1..24);

b[n_, p_, i_] := b[n, p, i] = If[n == 0, p!, If[i < 1, 0, b[n, p, i - 1] + a[i] b[n - i, p - 1, Min[n - i, i]]/(i - 1)!]];
a[n_] := If[n < 2, 1, b[n, n, n - 1]];
Array[a, 24] (* Jean-François Alcover, May 03 2020, after 2nd Maple program *)

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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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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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).

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A364207 Number of partitions of [n] such that the minimal element of each block is also its size.

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A327729 a(n) = Sum_{p} M(n-k; p_1-1, ..., p_k-1) * Product_{j=1..k} a(p_j), where p = (p_1, ..., p_k) ranges over all partitions of n into smaller parts (k is a partition length and M is a multinomial).

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