A179973 -id:A179973 - cp's OEIS frontend

A364128 Decimal expansion of a constant related to A053529 and A179973.

4, 4, 3, 2, 3, 8, 9, 5, 4, 7, 3, 0, 9, 2, 8, 5, 0, 9, 4, 0, 7, 7, 7, 5, 1, 2, 0, 7, 2, 8, 3, 3, 1, 8, 5, 1, 5, 0, 2, 0, 7, 2, 1, 9, 2, 4, 3, 9, 1, 5, 3, 0, 8, 7, 0, 7, 7, 6, 2, 9, 2, 8, 7, 8, 5, 3, 4, 5, 9, 1, 5, 9, 1, 4, 4, 7, 8, 7, 3, 5, 9, 3, 2, 5, 5, 7, 6, 1, 1, 6, 9, 2, 9, 1, 3, 8, 2, 8, 7, 1, 6, 4, 8, 5, 8, 8

Offset: 0

Alois P. Heinz, Jul 09 2023

The digits of this constant are also the final digits of A179973(n) and of the n-th partial sum of A053529 for n -> oo, read from right.

			0.443238954730928509407775120728331851502072192439153087... .

Alois P. Heinz, Table of n, a(n) for n = 0..1000
Index entries for sequences related to final digits of numbers

Cf. A004086, A025016, A053529, A055642, A179973.

Equals lim_{n->oo} A004086(A179973(n))/10^A055642(A179973(n)).

Equals lim_{n->oo} A004086(s(n))/10^A055642(s(n)) with s(n) = Sum_{j=0..n} A053529(j).

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

Alois P. Heinz, Jul 05 2023

nonn

The cycle lengths are distinct as a consequence of the definition.

			a(3) = 3: (123), (132), (1)(23).
a(4) = 8: (1234), (1243), (1324), (1342), (1423), (1432), (1)(234), (1)(243).

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

Cf. A000009, A007838, A032020, A179973, A321520, A326493, A364277, A364406.

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 *)

A327711 Sum of multinomials M(n-k; p_1-1, ..., p_k-1), where p = (p_1, ..., p_k) ranges over all partitions of n (k is a partition length).

Original entry on oeis.org

1, 1, 2, 3, 6, 10, 27, 55, 171, 475, 1555, 4915, 20023, 68243, 288024, 1213828, 5435935, 23966970, 121432923, 578757824, 3130381590, 16427772974, 91877826663, 519546134163, 3199523135912, 18868494152257, 120274458082095, 772954621249540, 5219747666882153

Offset: 0

Alois P. Heinz, Sep 22 2019

nonn

Number of partitions of [n] whose block sizes are nondecreasing when blocks are ordered by their minima and these minima are {1..k} (for some k <= n). a(5) = 10: 12345, 13|245, 14|235, 15|234, 1|2345, 1|24|35, 1|25|34, 1|2|345, 1|2|3|45, 1|2|3|4|5.

Alois P. Heinz, Table of n, a(n) for n = 0..635
Wikipedia, Multinomial coefficients
Wikipedia, Partition (number theory)

Cf. A005651, A179973, A326493, A327712, A327729.

with(combinat):
a:= n-> add(multinomial(n-nops(p), map(
    x-> x-1, p)[], 0), p=partition(n)):
seq(a(n), n=0..28);
# second Maple program:
b:= proc(n, i, p) option remember; `if`(n=0, p!, `if`(i<2, 0,
      b(n, i-1, p)) +b(n-i, min(n-i, i), p-1)/(i-1)!)
    end:
a:= n-> b(n$3):
seq(a(n), n=0..28);

b[n_, i_, p_] := b[n, i, p] = If[n == 0, p!, If[i < 2, 0, b[n, i - 1, p]] + b[n - i, Min[n - i, i], p - 1]/(i - 1)!];
a[n_] := b[n, n, n];
a /@ Range[0, 28] (* Jean-François Alcover, May 01 2020, from 2nd Maple program *)

A179972 Irregular table T(n,k) = A178886(n,k)/A048996(n,k) read by rows.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 6, 2, 2, 1, 1, 24, 6, 6, 2, 2, 1, 1, 120, 24, 24, 24, 6, 6, 6, 2, 2, 1, 1, 720, 120, 120, 120, 24, 24, 24, 24, 6, 6, 6, 2, 2, 1, 1, 5040, 720, 720, 720, 720, 120, 120, 120, 120, 120, 24, 24, 24, 24, 24, 6, 6, 6, 2, 2, 1, 1, 40320

Offset: 1

Alford Arnold, Aug 04 2010

Row n has A000041(n) terms.
Consider the five partitions of the number 4:
4 3+1 2+2 2+1+1 and 1+1+1+1
rewriting as 4000 3100 2200 2110 and 1111
then a(n) counts the ways that the zeros can be permuted:
6,2,2,1,1
agreeing with the factorial of the difference between A036042 and A036043.

			Row four of A178886 begins: 6 4 2 3 1
Row four of A048996 begins: 1 2 1 3 1
so,
Row four of A179972 begins: 6 2 2 1 1
Triangle T(n,k) begins:
    1;
    1,  1;
    2,  1,  1;
    6,  2,  2,  1, 1;
   24,  6,  6,  2, 2, 1, 1;
  120, 24, 24, 24, 6, 6, 6, 2, 2, 1, 1;
  ...

Cf. A178886, A048996, A036042, A036043, A179973 (row sums).

T(n,k) = ( A036042(n,k) - A036043(n,k))!.

T(n,k) = n!/A178888(n,k). - R. J. Mathar, Mar 03 2011

A179974 Triangle read by rows: T(n,k) = (n-A049085(n,k))! in columns 1<=k<=A000041(n), rows n>=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 2, 6, 1, 1, 2, 2, 6, 6, 24, 1, 1, 2, 6, 2, 6, 24, 6, 24, 24, 120, 1, 1, 2, 6, 2, 6, 24, 24, 6, 24, 120, 24, 120, 120, 720, 1, 1, 2, 6, 24, 2, 6, 24, 24, 120, 6, 24, 120, 120, 720, 24, 120, 720, 120, 720, 720, 5040, 1, 1, 2, 6, 24, 2, 6, 24, 120, 24, 120, 720, 6, 24, 120, 120, 720, 720, 24, 120, 720, 720, 5040, 120, 720, 5040

Offset: 1

Alford Arnold, Aug 05 2010

Since A049085 is a resortment of A036043 both A179972 and A179974 have row sums equal to A179973.

			Triangle begins
1;
1,1;
1,1,2;
1,1,2,2,6;
1,1,2,2,6,6,24;
1,1,2,6,2,6,24,6,24,24,120;
1,1,2,6,2,6,24,24,6,24,120,24,120,120,720;
1,1,2,6,24,2,6,24,24,120,6,24,120,120,720,24,120,720,120,720,720,5040;
1,1,2,6,24,2,6,24,120,24,120,720,6,24,120,120,720,720,24,120,720,720,5040,120,720,5040,720,5040,5040,40320,

Cf. A000041 (row lengths), A179973 (row sums), A036042, A049085 (max part).

cp's OEIS Frontend

A364128 Decimal expansion of a constant related to A053529 and A179973.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A362362 Number of permutations of [n] such that each cycle contains its length as an element.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A327711 Sum of multinomials M(n-k; p_1-1, ..., p_k-1), where p = (p_1, ..., p_k) ranges over all partitions of n (k is a partition length).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A179972 Irregular table T(n,k) = A178886(n,k)/A048996(n,k) read by rows.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

A179974 Triangle read by rows: T(n,k) = (n-A049085(n,k))! in columns 1<=k<=A000041(n), rows n>=1.

Views

Author

Keywords

Comments

Examples

Crossrefs