A001205 -id:A001205 - cp's OEIS frontend

A348070 Triangular array read by rows: T(n,k) is the number of undirected 2-regular labeled graphs whose largest connected component has exactly k nodes; n >= 1, 1 <= k <= n.

0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 0, 12, 0, 0, 10, 0, 0, 60, 0, 0, 0, 105, 0, 0, 360, 0, 0, 0, 315, 672, 0, 0, 2520, 0, 0, 280, 0, 4536, 5040, 0, 0, 20160, 0, 0, 0, 6300, 18144, 37800, 43200, 0, 0, 181440, 0, 0, 0, 51975, 55440, 332640, 356400, 415800, 0, 0, 1814400

Offset: 1

Steven Finch, Sep 27 2021

For the statistic "length of the smallest component", see A348071.

			Triangle begins:
  0;
  0,  0;
  0,  0,   1;
  0,  0,   0,   3;
  0,  0,   0,   0,   12;
  0,  0,  10,   0,    0,   60;
  0,  0,   0, 105,    0,    0,  360;
  0,  0,   0, 315,  672,    0,    0, 2520;
  0,  0, 280,   0, 4536, 5040,    0,    0, 20160;
...

Steven Finch, Permute, Graph, Map, Derange, arXiv:2111.05720 [math.CO], 2021.
D. Panario and B. Richmond, Exact largest and smallest size of components, Algorithmica, 31 (2001), 413-432.

Row sums give A001205, n >= 1.
Right border gives A001710.
Columns 1 and 2 each give A000004.
Cf. A348071.

T(n,n) = A001710(n-1) for n >= 2.

A348071 Triangular array read by rows: T(n,k) is the number of undirected 2-regular labeled graphs whose smallest connected component has exactly k nodes; n >= 1, 1 <= k <= n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 3, 0, 0, 0, 0, 12, 0, 0, 10, 0, 0, 60, 0, 0, 105, 0, 0, 0, 360, 0, 0, 672, 315, 0, 0, 0, 2520, 0, 0, 5320, 4536, 0, 0, 0, 0, 20160, 0, 0, 49500, 37800, 18144, 0, 0, 0, 0, 181440, 0, 0, 523215, 356400, 332640, 0, 0, 0, 0, 0, 1814400

Offset: 1

Steven Finch, Sep 27 2021

For the statistic "length of the largest component", see A348070.

			Triangle begins:
  0;
  0,  0;
  0,  0,    1;
  0,  0,    0,    3;
  0,  0,    0,    0,  12;
  0,  0,   10,    0,   0,  60;
  0,  0,  105,    0,   0,   0,  360;
  0,  0,  672,  315,   0,   0,    0, 2520;
  0,  0, 5320, 4536,   0,   0,    0,    0, 20160;
...

Steven Finch, Permute, Graph, Map, Derange, arXiv:2111.05720 [math.CO], 2021.
D. Panario and B. Richmond, Exact largest and smallest size of components, Algorithmica, 31 (2001), 413-432.

Row sums give A001205, n >= 1.
Right border gives A001710.
Columns 1 and 2 each give A000004.
Cf. A348070.

T(n,n) = A001710(n-1) for n >= 2.

A382134 Number of completely asymmetric matchings (not containing centered or coupled arcs) of [2n].

Original entry on oeis.org

1, 0, 0, 8, 48, 384, 4480, 59520, 897792, 15368192, 293769216, 6198589440, 143130972160, 3590253477888, 97214510235648, 2826205634330624, 87801981951344640, 2902989352269250560, 101776549707306237952, 3771425415371470405632, 147285455218020210180096

Offset: 0

R. J. Mathar, Mar 17 2025

nonn

Sergi Elizalde and Emeric Deutsch, The degree of asymmetry of a sequence, Enum. Combinat. Applic. 2 (2022) no 1 #S2R7, theorem 5.1

Cf. A000079, A001205, A047974, A053871.

g:= exp(-x-x^2)/sqrt(1-2*x) ;
seq( coeftayl(g,x=0,n)*n!,n=0..10) ;

E.g.f: exp(-x-x^2)/sqrt(1-2*x).
a(n) = 2^n * A001205(n).
D-finite with recurrence a(n) +2*(-n+1)*a(n-1) -4*(n-1)*(n-2)*a(n-3)=0.

A109542 a(n) = number of labeled 3-regular (trivalent) multi-graphs without self-loops on 2n vertices with a maximum of 2 edges between any pair of nodes. Also a(n) = number of labeled symmetric 2n X 2n matrices with {0,1,2}-entries with row sum equal to 3 for each row and trace 0.

Original entry on oeis.org

0, 7, 640, 170555, 94949400, 95830621425, 159062872168200, 404720953797785625

Offset: 1

Jeremy Gardiner, Aug 29 2005

			a(2)=7 because for 2*n=4 nodes there are 7 possible labeled graphs whose adjacency matrices are as follows:
0 2 1 0
2 0 0 1
1 0 0 2
0 1 2 0;
0 1 2 0
1 0 0 2
2 0 0 1
0 2 1 0;
0 2 0 1
2 0 1 0
0 1 0 2
1 0 2 0;
0 1 1 1
1 0 1 1
1 1 0 1
1 1 1 0;
0 0 2 1
0 0 1 2
2 1 0 0
1 2 0 0;
0 1 0 2
1 0 2 0
0 2 0 1
2 0 1 0;
0 0 1 2
0 0 2 1
1 2 0 0
2 1 0 0.

Cf. A001205, A002829, A108243.

a(5)-a(8) from Max Alekseyev, Aug 30 2005

A217763 Triangular array read by rows: T(n,k) is the number of simple labeled graphs on n nodes with unicyclic components having exactly k nodes with degree 1; n>=3, 0<=k<=n-3.

Original entry on oeis.org

1, 3, 12, 12, 90, 120, 70, 600, 1800, 1200, 465, 4725, 19530, 31500, 12600, 3507, 42168, 211680, 529200, 529200, 141120, 30016, 414288, 2451456, 7902720, 13124160, 8890560, 1693440, 286884, 4460760, 30413880, 117573120, 266716800, 312439680, 152409600, 21772800

Offset: 3

Geoffrey Critzer, Mar 23 2013

Column k=0 is A001205.

Row sums are A137916.

			  ....o-o..........o-o......
  ....| |..........|\ ......
  ....o-o..........o-o......
  T(4,0)=3 because the graph on the left has 4 nodes and 0 nodes with degree 1. It has 3 labelings.
  T(4,1)=12 because the graph on the right has 4 nodes and 1 node with degree 1.  It has 12 labelings.
1,
3,     12,
12,    90,     120,
70,    600,    1800,    1200,
465,   4725,   19530,   31500,   12600,
3507,  42168,  211680,  529200,  529200,   141120,
30016, 414288, 2451456, 7902720, 13124160, 8890560, 1693440.

nn=10;f[list_]:=Select[list,#>0&];t=Sum[Sum[n!/k! StirlingS2[n-1,n-k]y^k x^n/n!,{k,1,n}],{n,0,nn}];Map[Reverse,Map[f,Drop[Range[0,nn]!CoefficientList[Series[ Exp[Log[1/(1-t)]/2-t/2-t^2/4],{x,0,nn}],{x,y}],3]]]//Grid

exp(A(B(x,y)), where A(x) is e.g.f. for A137916 and B(x,y) is e.g.f. for A055302, gives T(n,n-k) (offset).

cp's OEIS Frontend

A348070 Triangular array read by rows: T(n,k) is the number of undirected 2-regular labeled graphs whose largest connected component has exactly k nodes; n >= 1, 1 <= k <= n.

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A348071 Triangular array read by rows: T(n,k) is the number of undirected 2-regular labeled graphs whose smallest connected component has exactly k nodes; n >= 1, 1 <= k <= n.

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A382134 Number of completely asymmetric matchings (not containing centered or coupled arcs) of [2n].

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Maple

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A109542 a(n) = number of labeled 3-regular (trivalent) multi-graphs without self-loops on 2n vertices with a maximum of 2 edges between any pair of nodes. Also a(n) = number of labeled symmetric 2n X 2n matrices with {0,1,2}-entries with row sum equal to 3 for each row and trace 0.

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A217763 Triangular array read by rows: T(n,k) is the number of simple labeled graphs on n nodes with unicyclic components having exactly k nodes with degree 1; n>=3, 0<=k<=n-3.

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Mathematica

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