A119741 -id:A119741 - cp's OEIS frontend

A268216 Duplicate of A119741.

2, 3, 6, 4, 12, 24, 5, 20, 60, 120, 6, 30, 120, 360, 720, 7, 42, 210, 840, 2520, 5040, 8, 56, 336, 1680, 6720, 20160, 40320, 9, 72, 504, 3024, 15120, 60480, 181440, 362880, 10, 90, 720, 5040, 30240, 151200, 604800, 1814400, 3628800

Offset: 2

dead

A173333 Triangle read by rows: T(n, k) = n! / k!, 1 <= k <= n.

Original entry on oeis.org

1, 2, 1, 6, 3, 1, 24, 12, 4, 1, 120, 60, 20, 5, 1, 720, 360, 120, 30, 6, 1, 5040, 2520, 840, 210, 42, 7, 1, 40320, 20160, 6720, 1680, 336, 56, 8, 1, 362880, 181440, 60480, 15120, 3024, 504, 72, 9, 1, 3628800, 1814400, 604800, 151200, 30240, 5040, 720, 90, 10, 1

Offset: 1

Reinhard Zumkeller, Feb 19 2010

From Wolfdieter Lang, Jun 27 2012: (Start)

T(n-1,k), k=1,...,n-1, gives the number of representative necklaces with n beads (C_N symmetry) of n+1-k distinct colors, say c[1],c[2],...,c[n-k+1], corresponding to the color signature determined by the partition k,1^(n-k) of n. The representative necklaces have k beads of color c[1]. E.g., n=4, k=2: partition 2,1,1, color signature (parts as exponents) c[1]c[1]c[2]c[3], 3=T(3,2) necklaces (write j for color c[j]): cyclic(1123), cyclic(1132) and cyclic(1213). See A212359 for the numbers for general partitions or color signatures. (End)

			Triangle starts:
n\k      1       2      3      4     5    6   7  8  9 10 ...
1        1
2        2       1
3        6       3      1
4       24      12      4      1
5      120      60     20      5     1
6      720     360    120     30     6    1
7     5040    2520    840    210    42    7   1
8    40320   20160   6720   1680   336   56   8  1
9   362880  181440  60480  15120  3024  504  72  9  1
10 3628800 1814400 604800 151200 30240 5040 720 90 10  1
... - _Wolfdieter Lang_, Jun 27 2012

Reinhard Zumkeller, Rows n = 1..150 of triangle, flattened
Index entries for sequences related to factorial numbers.

Row sums give A002627.
Central terms give A006963:
T(2*n-1,n) = A006963(n+1).
T(2*n,n) = A001813(n).
T(2*n,n+1) = A001761(n).
1 < k <= n: T(n,k) = T(n,k-1) / k.
1 <= k <= n: T(n+1,k) = A119741(n,n-k+1).
1 <= k <= n: T(n+1,k+1) = A162995(n,k).
T(n,1) = A000142(n).
T(n,2) = A001710(n) for n>1.
T(n,3) = A001715(n) for n>2.
T(n,4) = A001720(n) for n>3.
T(n,5) = A001725(n) for n>4.
T(n,6) = A001730(n) for n>5.
T(n,7) = A049388(n-7) for n>6.
T(n,8) = A049389(n-8) for n>7.
T(n,9) = A049398(n-9) for n>8.
T(n,10) = A051431(n) for n>9.
T(n,n-7) = A159083(n+1) for n>7.
T(n,n-6) = A053625(n+1) for n>6.
T(n,n-5) = A052787(n) for n>5.
T(n,n-4) = A052762(n) for n>4.
T(n,n-3) = A007531(n) for n>3.
T(n,n-2) = A002378(n-1) for n>2.
T(n,n-1) = A000027(n) for n>1.
T(n,n) = A000012(n).
Cf. A138533, A002627.

a173333 n k = a173333_tabl !! (n-1) !! (k-1)
a173333_row n = a173333_tabl !! (n-1)
a173333_tabl = map fst $ iterate f ([1], 2)
   where f (row, i) = (map (* i) row ++ [1], i + 1)
-- Reinhard Zumkeller, Jul 04 2012

Table[n!/k!, {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Mar 01 2019 *)

E.g.f.: (exp(x*y) - 1)/(x*(1 - y)). - Olivier Gérard, Jul 07 2011

T(n,k) = A094587(n,k), 1 <= k <= n. - Reinhard Zumkeller, Jul 05 2012

A038156 a(n) = n! * Sum_{k=1..n-1} 1/k!.

Original entry on oeis.org

0, 0, 2, 9, 40, 205, 1236, 8659, 69280, 623529, 6235300, 68588311, 823059744, 10699776685, 149796873604, 2246953104075, 35951249665216, 611171244308689, 11001082397556420, 209020565553571999, 4180411311071440000, 87788637532500240021, 1931350025715005280484

Offset: 0

N. J. A. Sloane

Related to number of operations of addition and multiplication to evaluate a determinant of order n by cofactor expansion - see A026243.
Also number of operations needed to create all permutations of n distinct elements using Algorithm L (lexicographic permutation generation) from Knuth's The Art of Computer Programming, Vol. 4, chapter 7.2.1.2. Sequence gives number of comparisons required to find j in step L2 (see answer to exercise 5). - Hugo Pfoertner, Jan 24 2003
For n>1, the number of possible ballots where there are n candidates and voters may identify their top m most preferred ones, where 0 < m < n. - Shaye Horwitz, Jun 28 2011
For n > 1, a(n) is the expected number of comparisons required to sort a random list of n distinct elements using the "bogosort" algorithm. - Andrew Slattery, Jun 02 2022
The number of permutations of all proper nonempty subsets of an n element set. - P. Christopher Staecker, May 09 2024

			a(2) = floor((2.718... - 1)*2) - 1 = 3 - 1 = 2,
a(3) = floor((2.718... - 1)*6) - 1 = 10 - 1 = 9.

D. E. Knuth: The Art of Computer Programming, Volume 4, Fascicle 2. Generating All Tuples and Permutations, Addison-Wesley, 2005.

Georg Fischer, Table of n, a(n) for n = 0..200 [first 28 terms from _Vincenzo Librandi_]
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 836
G. A. Kamel, Partial Chain Topologies on Finite Sets, Computational and Applied Mathematics Journal. Vol. 1, No. 4, 2015, pp. 174-179.
Hugo Pfoertner, FORTRAN implementation of Knuth's Algorithm L for lexicographic permutation generation.
Wikipedia, Bogosort
Index entries for sequences related to factorial numbers

Cf. A000522, A007526, A038155, A056542, A079884, A079750.

Row sums of A119741.

a:= proc(n) option remember;
      `if`(n<2, 0, a(n-1)*n+n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Apr 11 2020

a=1; Join[{0},Table[a=(a-1)*(n+1);Abs[a],{n,0,60}]] (* Vladimir Joseph Stephan Orlovsky, Nov 20 2009; 0 prefixed by _Georg Fischer Apr 11 2020 *)
Join[{0},FoldList[#1*#2 + #2 + #1 + 1 &, 0, Range@ 20]] (* Robert G. Wilson v, Feb 21 2015 *)

a(n)=floor((exp(1)-1)*n!-1) \\ Charles R Greathouse IV, Jun 29 2011

a(n)=(expm1(1)*n!-1)\1 \\ Charles R Greathouse IV, Jan 28 2014

a(n) = floor((e-1)*n!) - 1.
a(0) = a(1) = 0, a(n) = n*(a(n-1) + 1) for n>1. - Philippe Deléham, Oct 16 2009
E.g.f.: (exp(x) - 1)*x/(1 - x). - Ilya Gutkovskiy, Jan 26 2017
a(n) = A002627(n)-1, n>=1. - R. J. Mathar, Jan 03 2018
a(n) = A000522(n)-n!-1, n>=1. - P. Christopher Staecker, May 09 2024

a(28) ff. corrected by Georg Fischer, Apr 11 2020

A268217 Triangle read by rows: T(n,k) (n>=3, k=3..n) is the number of topologies t on n points having exactly k open sets such that t contains exactly one open set of size m for each m in {0,2,3,4,...,s,n} where s is the size of the largest proper open set in t.

Original entry on oeis.org

3, 6, 12, 10, 30, 60, 15, 60, 180, 360, 21, 105, 420, 1260, 2520, 28, 168, 840, 3360, 10080, 20160, 36, 252, 1512, 7560, 30240, 90720, 181440, 45, 360, 2520, 15120, 75600, 302400, 907200, 1814400, 55, 495, 3960, 27720, 166320, 831600, 3326400, 9979200, 19958400

Offset: 3

N. J. A. Sloane, Jan 29 2016

When two leading 0's are added and last element repeated, rows give the coefficients of the path polynomials of the complete graph K_n. - Eric W. Weisstein, Jun 04 2017

			Triangle begins:
   3;
   6,  12;
  10,  30,   60;
  15,  60,  180,   360;
  21, 105,  420,  1260,   2520;
  28, 168,  840,  3360,  10080,  20160;
  36, 252, 1512,  7560,  30240,  90720,  181440;
  45, 360, 2520, 15120,  75600, 302400,  907200, 1814400;
  ...

Andrew Howroyd, Table of n, a(n) for n = 3..1277 (first 50 rows)
G. A. Kamel, Partial Chain Topologies on Finite Sets, Computational and Applied Mathematics Journal. Vol. 1, No. 4, 2015, pp. 174-179.
Eric Weisstein's World of Mathematics, Complete Graph.
Eric Weisstein's World of Mathematics, Graph Path.

Row sums give A038158.
Triangles in this series: A119741, A268217, A268221, A268222, A268223.
Cf. A282507.

i = 2; Table[Table[Binomial[n, i] FactorialPower[n - i, k], {k, 0, n - i - 1}], {n, 2, 9}] // Grid (* Geoffrey Critzer, Feb 19 2017 *)
CoefficientList[Table[-(1/2) (n - 1) n x^(n - 2) (Gamma[n - 1] - E^(1/x) Gamma[n - 1, 1/x]), {n, 3, 10}] // FunctionExpand, x] // Flatten (* Eric W. Weisstein, Jun 04 2017 *)

Title clarified by Geoffrey Critzer, Feb 19 2017

Corrected and extended by Andrew Howroyd, Aug 09 2025

A268221 Triangle read by rows: T(n,k) (n>=4, k=3..n-1) is the number of topologies t on n points having exactly k open sets such that t contains exactly one open set of size m for each m in {0,3,4,5,...,s,n} where s is the size of the largest proper open set in t.

Original entry on oeis.org

4, 10, 20, 20, 60, 120, 35, 140, 420, 840, 56, 280, 1120, 3360, 6720, 84, 504, 2520, 10080, 30240, 60480, 120, 840, 5040, 25200, 100800, 302400, 604800, 165, 1320, 9240, 55440, 277200, 1108800, 3326400, 6652800, 220, 1980, 15840, 110880, 665280, 3326400, 13305600, 39916800, 79833600

Offset: 4

N. J. A. Sloane, Jan 30 2016

			Triangle begins:
    4;
   10,  20;
   20,  60,  120;
   35, 140,  420,   840;
   56, 280, 1120,  3360,   6720;
   84, 504, 2520, 10080,  30240,  60480;
  120, 840, 5040, 25200, 100800, 302400, 604800;
  ...

Andrew Howroyd, Table of n, a(n) for n = 4..1278 (first 50 rows)
G. A. Kamel, Partial Chain Topologies on Finite Sets, Computational and Applied Mathematics Journal. Vol. 1, No. 4, 2015, pp. 174-179.

Row sums give A268218.
Triangles in this series: A119741, A268217, A268221, A268222, A268223.
Cf. A282507.

i = 3; Table[Table[Binomial[n, i] FactorialPower[n - i, k], {k, 0, n - i - 1}], {n, 2, 12}] // Grid (* Geoffrey Critzer, Feb 19 2017 *)

Title clarified and more terms added by Geoffrey Critzer, Feb 19 2017

A268222 Triangle read by rows: T(n,k) (n>=5, k=3..n-2) is the number of topologies t on n points having exactly k open sets such that t contains exactly one open set of size m for each m in {0,4,5,6,...,s,n} where s is the size of the largest proper open set in t.

Original entry on oeis.org

5, 15, 30, 35, 105, 210, 70, 280, 840, 1680, 126, 630, 2520, 7560, 15120, 210, 1260, 6300, 25200, 75600, 151200, 330, 2310, 13860, 69300, 277200, 831600, 1663200, 495, 3960, 27720, 166320, 831600, 3326400, 9979200, 19958400, 715, 6435, 51480, 360360, 2162160, 10810800, 43243200, 129729600, 259459200

Offset: 5

N. J. A. Sloane, Jan 30 2016

			Triangle begins:
    5;
   15,   30;
   35,  105,  210;
   70,  280,  840,  1680;
  126,  630, 2520,  7560, 15120;
  210, 1260, 6300, 25200, 75600, 151200;
...

Andrew Howroyd, Table of n, a(n) for n = 5..1279 (first 50 rows)
G. A. Kamel, Partial Chain Topologies on Finite Sets, Computational and Applied Mathematics Journal. Vol. 1, No. 4, 2015, pp. 174-179.

Row sums give A268219.
Triangles in this series: A119741, A268217, A268221, A268222, A268223.
Cf. A282507.

i = 4; Table[Table[Binomial[n, i] FactorialPower[n - i, k], {k, 0,
n - i - 1}], {n, 2, 12}] // Grid (* Geoffrey Critzer, Feb 19 2017 *)

Title clarified and more terms added by Geoffrey Critzer, Feb 19 2017

Missing a(19) inserted and a(41) onwards from Andrew Howroyd, Aug 10 2025

A268223 Triangle read by rows: T(n,k) (n>=6, k=3..n-3) is the number of topologies t on n points having exactly k open sets such that t contains exactly one open set of size m for each m in {0,5,6,7,...,s,n} where s is the size of the largest proper open set in t.

Original entry on oeis.org

6, 21, 42, 56, 168, 336, 126, 504, 1512, 3024, 252, 1260, 5040, 15120, 30240, 462, 2772, 13860, 55440, 166320, 332640, 792, 5544, 33264, 166320, 665280, 1995840, 3991680, 1287, 10296, 72072, 432432, 2162160, 8648640, 25945920, 51891840

Offset: 6

N. J. A. Sloane, Jan 30 2016

			Triangle begins:
    6;
   21,   42;
   56,  168,  336;
  126,  504, 1512,  3024;
  252, 1260, 5040, 15120, 30240;
  ...

Andrew Howroyd, Table of n, a(n) for n = 6..1280 (first 50 rows)
G. A. Kamel, Partial Chain Topologies on Finite Sets, Computational and Applied Mathematics Journal. Vol. 1, No. 4, 2015, pp. 174-179.

Row sums give A268219.

Triangles in this series: A119741, A268217, A268221, A268222, A268223.

i = 5; Table[ Table[Binomial[n, i] FactorialPower[n - i, k], {k, 0,
n - i - 1}], {n, 2, 13}] // Grid (* Geoffrey Critzer, Feb 19 2017 *)

Title clarified and more terms added by Geoffrey Critzer, Feb 19 2017

A282507 Triangular array read by rows. T(n,k) is the number of chain topologies on an n-set with exactly k open sets where one of the open sets is a single point set, n >= 2, 3 <= k <= n+1.

Original entry on oeis.org

2, 3, 6, 4, 24, 24, 5, 70, 180, 120, 6, 180, 900, 1440, 720, 7, 434, 3780, 10920, 12600, 5040, 8, 1008, 14448, 67200, 134400, 120960, 40320, 9, 2286, 52164, 367416, 1134000, 1723680, 1270080, 362880, 10, 5100, 181500, 1864800, 8341200, 19051200, 23284800, 14515200, 3628800

Offset: 2

Geoffrey Critzer, Feb 16 2017

A chain topology is a topology that can be totally ordered by inclusion.

			Triangle begins:
  2;
  3,   6;
  4,  24,  24;
  5,  70, 180,  120;
  6, 180, 900, 1440, 720;
  ...

Michael De Vlieger, Table of n, a(n) for n = 2..1226 (rows 2..50, flattened)
Loïc Foissy, Hopf algebraic structures on hypergraphs and multi-complexes, arXiv:2304.00810 [math.CO], 2023.

Cf. A119741 where the topologies are further restricted.
Row sums = A052882.
Cf. A019538.

nn = 10; Map[Select[#, # > 0 &] &, Drop[Range[0, nn]! CoefficientList[Series[x/(1 - y (Exp[x] - 1)), {x, 0, nn}], {x, y}], 2]] // Grid

E.g.f.: y^2*x/(1 - y*(exp(x) - 1)). Generally for chain topologies where the smallest nonempty open set has size m: (x^m/m!) * y^2/(1 - y*(exp(x) - 1)).

A conjecture I made to Loic Foissy, who replied: sequence T(n,k) counts surjective maps [n]->> [k] such that k is obtained exactly once, whereas sequence A019538 b(n,k) counts surjective maps [n]->> [k]. To construct an element for T(n,k), you may choose the element of [n] giving k (n choices), then a surjection from the n-1 remaining elements to [k-1] (b(n-1,k-1) choices). This gives T(n,k) = n * b(n-1,k-1), if k,n>1. - Tom Copeland, Nov 10 2023 [So it is now a theorem, not a conjecture, right? - N. J. A. Sloane, Dec 23 2023]

cp's OEIS Frontend

A268216 Duplicate of A119741.

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A173333 Triangle read by rows: T(n, k) = n! / k!, 1 <= k <= n.

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A038156 a(n) = n! * Sum_{k=1..n-1} 1/k!.

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A268217 Triangle read by rows: T(n,k) (n>=3, k=3..n) is the number of topologies t on n points having exactly k open sets such that t contains exactly one open set of size m for each m in {0,2,3,4,...,s,n} where s is the size of the largest proper open set in t.

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A268221 Triangle read by rows: T(n,k) (n>=4, k=3..n-1) is the number of topologies t on n points having exactly k open sets such that t contains exactly one open set of size m for each m in {0,3,4,5,...,s,n} where s is the size of the largest proper open set in t.

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A268222 Triangle read by rows: T(n,k) (n>=5, k=3..n-2) is the number of topologies t on n points having exactly k open sets such that t contains exactly one open set of size m for each m in {0,4,5,6,...,s,n} where s is the size of the largest proper open set in t.

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A268223 Triangle read by rows: T(n,k) (n>=6, k=3..n-3) is the number of topologies t on n points having exactly k open sets such that t contains exactly one open set of size m for each m in {0,5,6,7,...,s,n} where s is the size of the largest proper open set in t.

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A282507 Triangular array read by rows. T(n,k) is the number of chain topologies on an n-set with exactly k open sets where one of the open sets is a single point set, n >= 2, 3 <= k <= n+1.

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