A058716 -id:A058716 - cp's OEIS frontend

A000065 -1 + number of partitions of n.

0, 0, 1, 2, 4, 6, 10, 14, 21, 29, 41, 55, 76, 100, 134, 175, 230, 296, 384, 489, 626, 791, 1001, 1254, 1574, 1957, 2435, 3009, 3717, 4564, 5603, 6841, 8348, 10142, 12309, 14882, 17976, 21636, 26014, 31184, 37337, 44582, 53173, 63260, 75174, 89133, 105557, 124753

Offset: 0

N. J. A. Sloane

a(n+1) is the number of noncongruent n-dimensional integer-sided simplices with diameter n. - Sascha Kurz, Jul 26 2004
Also, the number of partitions of n into parts each less than n.
Also, the number of distinct types of equation which can be derived from the equation [n,0,0] not including itself. (Ince)
Also, the number of rooted trees on n+1 nodes with height exactly 2.
Also, the number of partitions (of any positive integer) whose sum + length is <= n. Example: a(5) = 6 counts 4, 3, 21, 2, 11, 1. Proof: Given a partition of n other than the all 1s partition, subtract 1 from each part and then drop the zeros. This is a bijection to the partitions with sum + length <= n. - David Callan, Nov 29 2007
Number of graphs with n vertices of treewidth n-2. Reason: The complement of a graph with n vertices and treewidth >= n-2 cannot have P3 or K3 as a subgraph (Chlebı́ková 2002, Theorem 10), so every component of it is a star. - Martín Muñoz, Dec 31 2023

			G.f. = x^2 + 2*x^3 + 4*x^4 + 6*x^5 + 10*x^6 + 14*x^7 + 21*x^8 + 29*x^9 + ...

E. L. Ince, Ordinary Differential Equations, Dover Publications, New York, 1944, p. 498; MR0010757.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000 (first 199 terms from N. J. A. Sloane)
Janka Chlebı́ková, The Structure of Obstructions to Treewidth and Pathwidth. Disc. Appl. Math., Vol. 120, no. 1-3 (2002), 61-71.
Harmandeep Kaur and Muhammad Asif Rana, Partitions with unique largest part and their generating functions, arXiv:2506.11447 [math.CO], 2025.
Vladimir Modrak and David Marton, Development of Metrics and a Complexity Scale for the Topology of Assembly Supply Chains, Entropy 2013, 15, 4285-4299
J. Riordan, Enumeration of trees by height and diameter, IBM J. Res. Dev. 4 (1960), 473-478.
J. Riordan, The enumeration of trees by height and diameter, IBM Journal 4 (1960), 473-478. (Annotated scanned copy)

A000041 - 1. A column of A058716. A diagonal of A263294.

Column h=2 of A034781.

[NumberOfPartitions(n)-1: n in [0..50]]; // Vincenzo Librandi, Aug 25 2013

with (combstruct):ZL:=proc(m) local i; [T0,{seq(T.i=Prod(Z,Set(T.(i+1))),i=0..m-1), T.m=Z}, unlabeled] end:A:=n -> count(ZL(2),size=n)-count(ZL(1),size=n): seq(A(n),n=1..46); # Zerinvary Lajos, Dec 05 2007
ZL :=[S, {S = Set(Cycle(Z),1 < card)}, unlabelled]: seq(combstruct[count](ZL, size=n), n=0..45); # Zerinvary Lajos, Mar 25 2008

nn=40;CoefficientList[Series[Product[1/(1-x^i),{i,1,nn}]-1/(1-x),{x,0,nn}],x]  (* Geoffrey Critzer, Oct 28 2012 *)
PartitionsP[Range[0,50]]-1 (* Harvey P. Dale, Aug 24 2013 *)

{a(n) = if( n<0, 0, polcoeff( 1 / eta(x + x*O(x^n)), n) - 1)};

PARI
```
{a(n) = if( n<0, 0, numbpart(n) - 1)};
```

a(n) = A026820(n,n-1) for n>1. - Reinhard Zumkeller, Jan 21 2010
G.f.: x*G(0)/(x-1) where G(k) = 1 - 1/(1-x^(k+1))/(1-x/(x-1/G(k+1) )); (recursively defined continued fraction). - Sergei N. Gladkovskii, Jan 23 2013
G.f.: Sum_{k>=2} x^k / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Sep 07 2021

A058720 Triangle T(n,k) giving the number of simple matroids of rank k on n labeled points (n >= 2, 2 <= k <= n).

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 31, 16, 1, 1, 352, 337, 42, 1, 1, 8389, 18700, 2570, 99, 1, 1, 433038, 7642631, 907647, 16865, 219, 1

Offset: 2

N. J. A. Sloane, Dec 31 2000

			Triangle T(n,k) (with rows n >= 2 and columns k >= 2) begins as follows:
  1;
  1,      1;
  1,      5,       1;
  1,     31,      16,      1;
  1,    352,     337,     42,     1;
  1,   8389,   18700,   2570,    99,   1;
  1, 433038, 7642631, 907647, 16865, 219, 1;
  ...

Mohamed Barakat, Reimer Behrends, Christopher Jefferson, Lukas Kühne, and Martin Leuner, On the generation of rank 3 simple matroids with an application to Terao's freeness conjecture, arXiv:1907.01073 [math.CO], 2019.
W. M. B. Dukes, Tables of matroids.
W. M. B. Dukes, Counting and Probability in Matroid Theory, Ph.D. Thesis, Trinity College, Dublin, 2000.
W. M. B. Dukes, The number of matroids on a finite set, arXiv:math/0411557 [math.CO], 2004.
W. M. B. Dukes, On the number of matroids on a finite set, Séminaire Lotharingien de Combinatoire 51 (2004), Article B51g. [See p. 11.]
Index entries for sequences related to matroids

Cf. A000110 (Bell numbers), A002662, A058710, A058711, A058716, A058730, A100728.
Row sums give A058721.
Columns include (truncated versions of) A000012 (k=2), (A056642)+1 (k=3), A058722 (k=4).

From Petros Hadjicostas, Oct 09 2019: (Start)
T(n, n-1) = 2^n - 1 - binomial(n+1,2) = A002662(n) for n >= 2. [Dukes (2004), Lemma 2.2(i).]
T(n, n-2) = A100728(n) = A000110(n+1) + binomial(n+3,4) + 2*binomial(n+1,4) - 2^n - 2^(n-1)*binomial(n+1,2). [Dukes (2004), Lemma 2.2(iii).]
(End)

A058718 Number of nonisomorphic loopless matroids on n unlabeled points.

Original entry on oeis.org

1, 1, 2, 4, 9, 21, 60, 208, 1418, 381448

Offset: 0

N. J. A. Sloane, Dec 31 2000

W. M. B. Dukes, Tables of matroids.
W. M. B. Dukes, Counting and Probability in Matroid Theory, Ph.D. Thesis, Trinity College, Dublin, 2000.
W. M. B. Dukes, The number of matroids on a finite set, arXiv:math/0411557 [math.CO], 2004.
W. M. B. Dukes, On the number of matroids on a finite set, Séminaire Lotharingien de Combinatoire 51 (2004), Article B51g.
Dillon Mayhew and Gordon F. Royle, Matroids with nine elements, arXiv:math/0702316 [math.CO], 2007.
Dillon Mayhew and Gordon F. Royle, Matroids with nine elements, J. Combin. Theory Ser. B 98(2) (2008), 415-431.
Gordon Royle and Dillon Mayhew, 9-element matroids.
Index entries for sequences related to matroids

Row sums of A058716.

a(9) from Gordon Royle, Dec 23 2006

A058719 Number of nonisomorphic loopless matroids of rank 3 on n labeled points.

Original entry on oeis.org

1, 3, 9, 25, 70, 217, 950, 8762, 288454

Offset: 3

N. J. A. Sloane, Dec 31 2000

nonn

A. Evnin, An Elementary Introduction to Matroids, Mathematical Education, 2(33), 2005, 2-33.

W. M. B. Dukes, Tables of matroids
W. M. B. Dukes, Counting and Probability in Matroid Theory, Ph.D. Thesis, Trinity College, Dublin, 2000.
A. Evnin, An elementary Introduction to Matroids, Mathematical Education 2 (33) (2005), 2-33
Index entries for sequences related to matroids

A diagonal of A058716.

a(9)-a(11) Alexander Evnin, Dec 26 2008

A058717 Triangle T(n,k) giving number of nonisomorphic loopless matroids of rank k on n labeled points (n >= 1, 1<=k<=n).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 6, 9, 4, 1, 1, 10, 25, 18, 5, 1, 1, 14, 70, 85, 31, 6, 1, 1, 21, 217, 832, 288, 51, 7, 1

Offset: 1

N. J. A. Sloane, Dec 31 2000

			1;
1,  1;
1,  2,   1;
1,  4,   3,   1;
1,  6,   9,   4,   1;
1, 10,  25,  18,   5,  1;
1, 14,  70,  85,  31,  6, 1;
1, 21, 217, 832, 288, 51, 7, 1;
...

W. M. B. Dukes, Tables of matroids
W. M. B. Dukes, Counting and Probability in Matroid Theory, Ph.D. Thesis, Trinity College, Dublin, 2000.
Index entries for sequences related to matroids
W. M. B. Dukes, On the number of matroids on a finite set

Cf. A058716 (same except for border), A058710, A058711.

Row sums give A058718. Diagonals give A000065, A058719.

Corrected and extended by Jean-François Alcover, Oct 21 2013

Reverted to original data by Jean-François Alcover, Aug 17 2022

cp's OEIS Frontend

A000065 -1 + number of partitions of n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Formula

A058720 Triangle T(n,k) giving the number of simple matroids of rank k on n labeled points (n >= 2, 2 <= k <= n).

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A058718 Number of nonisomorphic loopless matroids on n unlabeled points.

Views

Author

Keywords

Links

Crossrefs

Extensions

A058719 Number of nonisomorphic loopless matroids of rank 3 on n labeled points.

Views

Author

Keywords

References

Links

Crossrefs

Extensions

A058717 Triangle T(n,k) giving number of nonisomorphic loopless matroids of rank k on n labeled points (n >= 1, 1<=k<=n).

Views

Author

Keywords

Examples

Links

Crossrefs

Extensions