A324166 -id:A324166 - cp's OEIS frontend

A000699 Number of irreducible chord diagrams with 2n nodes.

1, 1, 1, 4, 27, 248, 2830, 38232, 593859, 10401712, 202601898, 4342263000, 101551822350, 2573779506192, 70282204726396, 2057490936366320, 64291032462761955, 2136017303903513184, 75197869250518812754, 2796475872605709079512, 109549714522464120960474, 4509302910783496963256400, 194584224274515194731540740

Offset: 0

N. J. A. Sloane

Perturbation expansion in quantum field theory: spinor case in 4 spacetime dimensions.
a(n)*2^(-n) is the coefficient of the x^(2*n-1) term in the series reversal of the asymptotic expansion of 2*DawsonF(x) = sqrt(Pi)*exp(-x^2)*erfi(x) for x -> inf. - Vladimir Reshetnikov, Apr 23 2016
The September 2018 talk by Noam Zeilberger (see link to video) connects three topics (planar maps, Tamari lattices, lambda calculus) and eight sequences: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827. - N. J. A. Sloane, Sep 17 2018
A set partition is topologically connected if the graph whose vertices are the blocks and whose edges are crossing pairs of blocks is connected, where two blocks cross each other if they are of the form {{...x...y...},{...z...t...}} for some x < z < y < t or z < x < t < y. Then a(n) is the number of topologically connected 2-uniform set partitions of {1...2n}. See my links for examples. - Gus Wiseman, Feb 23 2019
From Julien Courtiel, Oct 09 2024: (Start)
a(n) is the number of rooted bridgeless combinatorial maps with n edges (genus is not fixed). A map is bridgeless if it has no edge whose removal disconnects the graph. For example, for n=2, there are 4 bridgeless maps with 2 edges: 2 planar maps with 1 vertex (either two consecutive loops, or two nested loops), 1 toric map with 1 vertex, and 1 planar map with 2 vertices connected by a double edge.
Also, a(n) is the number of trees with n edges equipped with a binary tubing. A tube is a connected subgraph. A binary tubing of a tree is a nested set collection S of tubes such that 1. S contains the tube of all vertices 2. Every tube of S is either reduced to one vertex, or it can be can partitioned by 2 tubes of S.
(End)

			a(31)=627625976637472254550352492162870816129760 was computed using Kreimer's Hopf algebra of rooted trees. It subsumes 2.6*10^21 terms in quantum field theory.
G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 27*x^4 + 248*x^5 + 2830*x^6 +...
where d/dx (A(x) - 1)^2/x = 1 + 4*x + 27*x^2 + 248*x^3 + 2830*x^4 +...

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).

Alois P. Heinz, Table of n, a(n) for n = 0..404 (terms up to n=100 from T. D. Noe)
Serban T. Belinschi, Marek Bozejko, Franz Lehner, and Roland Speicher, The normal distribution is "boxplus"-infinitely divisible, arXiv:0910.4263 [math.OA], 2009-2010.
Daniel J. Bernstein, S. Engels, T. Lange, R. Niederhagen, et al., Faster elliptic-curve discrete logarithms on FPGAs, Preprint, 2016.
Michael Borinsky, Generating asymptotics for factorially divergent sequences, arXiv preprint arXiv:1603.01236 [math.CO], 2016.
Michael Borinsky, Generating asymptotics for factorially divergent sequences, El. J. Combin. 25 (2018) P4.1, Sect 7.1.
Michael Borinsky, Gerald V. Dunne, and Karen Yeats, Tree-tubings and the combinatorics of resurgent Dyson-Schwinger equations, arXiv:2408.15883 [math-ph], 2024. See p. 9.
David J. Broadhurst and Dirk Kreimer, Combinatoric explosion of renormalization ..., arXiv:hep-th/9912093, 1999, 2000.
David J. Broadhurst and Dirk Kreimer, Combinatoric explosion of renormalization tamed by Hopf algebra: 30-loop Pad-Borel resummation, Phys. Lett. B 475 (2000), 63-70.
Christian Brouder, On the trees of quantum fields, arXiv:hep-th/9906111, 1999, p. 6.
Jonathan Burns, Assembly Graph Words - Single Transverse Component (Counts).
Jonathan Burns, Egor Dolzhenko, Natasa Jonoska, Tilahun Muche and Masahico Saito, Four-Regular Graphs with Rigid Vertices Associated to DNA Recombination, May 23, 2011.
Jonathan Burns and Tilahun Muche, Counting Irreducible Double Occurrence Words, arXiv preprint arXiv:1105.2926 [math.CO], 2011.
Julien Courtiel, Karen Yeats, and Noam Zeilberger, Connected chord diagrams and bridgeless maps, arXiv:1611.04611 [math.CO], 2016.
Serge Dulucq, Etude combinatoire de problèmes d'énumération, d'algorithmique sur les arbres et de codage par des mots, a thesis presented to l'Université de Bordeaux I, 1987. (Annotated scanned copy)
Philippe Flajolet and Marc Noy, Analytic Combinatorics of Chord Diagrams, in: Formal power series and algebraic combinatorics (FPSAC '00) Moscow, 2000, pp. 191-201.
Zhen Huang, Denis Golež, Hugo U. R. Strand, and Jason Kaye, Automated evaluation of imaginary time strong coupling diagrams by sum-of-exponentials hybridization fitting, arXiv:2503.19727 [cond-mat.str-el], 2025. See p. 21.
Martin Klazar, Non-P-recursiveness of numbers of matchings or linear chord diagrams with many crossings, Advances in Appl. Math., Vol. 30 (2003), pp. 126-136.
Martin Klazar, Counting even and odd partitions, Amer. Math. Monthly, 110 (No. 6, 2003), 527-532.
Florian Kogelbauer and Ilya Karlin, On the Relation of Exact Hydrodynamics to the Chapman-Enskog Series, arXiv:2506.17441 [math-ph], 2025. See pp. 4-5.
Ali Assem Mahmoud, On the Asymptotics of Connected Chord Diagrams, University of Waterloo (Ontario, Canada 2019).
Ali Assem Mahmoud, An Asymptotic Expansion for the Number of 2-Connected Chord Diagrams, arXiv:2009.12688 [math.CO], 2020.
Ali Assem Mahmoud, Chord Diagrams and the Asymptotic Analysis of QED-type Theories, arXiv:2011.04291 [hep-th], 2020.
Ali Assem Mahmoud, An asymptotic expansion for the number of two-connected chord diagrams, J. Math. Phys. (2023) Vol. 64, 122301. See Example 2.1.
Ali Assem Mahmoud and Karen Yeats, Connected Chord Diagrams and the Combinatorics of Asymptotic Expansions, arXiv:2010.06550 [math.CO], 2020.
Nicolas Marie and Karen Yeats, A chord diagram expansion coming from some Dyson-Schwinger equations, arXiv:1210.5457 [math.CO], 2012, Section 4.1.
Albert Nijenhuis and Herbert S. Wilf, The enumeration of connected graphs and linked diagrams, J. Combin. Theory Ser. A 27 (1979), no. 3, 356--359. MR0555804 (82b:05074).
Vincent Pilaud and Juanjo Rué, Analytic combinatorics of chord and hyperchord diagrams with k crossings, arXiv preprint arXiv:1307.6440 [math.CO], 2013.
J. Riordan, Letter, Jul 06 1978
Einar A. Rødland, Pseudoknots in RNA Secondary Structures: Representation, Enumeration, and Prevalence, J. Comput. Biology, Vol 13, No 6 (2006), 1197-1213. (see equation 10)
R. R. Stein, On a class of linked diagrams, I. Enumeration, J. Combin. Theory, A 24 (1978), 357-366.
R. R. Stein and C. J. Everett, On a class of linked diagrams, II. Asymptotics, Discrete Math., 21 (1978), 309-318.
Jacques Touchard, Sur un problème de configurations et sur les fractions continues, Canad. J. Math., 4 (1952), 2-25.
Jacques Touchard, Sur un problème de configurations et sur les fractions continues, Canad. J. Math., 4 (1952), 2-25. [Annotated, corrected, scanned copy]
T. R. S. Walsh and A. B. Lehman, Counting rooted maps by genus. III: Nonseparable maps, J. Combinatorial Theory Ser. B 18 (1975), 222-259 (nonseparable integer systems on n pairs). (Give an incorrect a(6)=2720.)
Gus Wiseman, The a(4) = 27 connected chord diagrams.
Gus Wiseman, The a(5) = 248 connected chord diagrams.
Gus Wiseman, Constructive Mathematica program for A000699.
Noam Zeilberger, A theory of linear typings as flows on 3-valent graphs, arXiv:1804.10540 [cs.LO], 2018.
Noam Zeilberger, A Sequent Calculus for a Semi-Associative Law, arXiv:1803.10080 [math.LO], 2018-2019 (A revised version of a 2017 conference paper).
Noam Zeilberger, A proof-theoretic analysis of the rotation lattice of binary trees, Part 1 (video), Rutgers Experimental Math Seminar, Sep 13 2018. Part 2 is vimeo.com/289910554.

Sequences mentioned in the Noam Zeilberger 2018 video: A000168, A000260, A000309, A000698, A000699, A002005, A062980, A267827.
Cf. A004300, A051862, A212273. Column sums of A232223. First column of A322402.
Cf. A007297, A016098, A099947, A136653, A268815, A306438, A324166, A324172, A324173, A324327.

A000699 := proc(n)
    option remember;
    if n <= 1 then
        1;
    else
        add((2*i-1)*procname(i)*procname(n-i),i=1..n-1) ;
    end if;
end proc:
seq(A000699(n),n=0..30) ; # R. J. Mathar, Jun 12 2018

terms = 22; A[] = 0; Do[A[x] = x + x^2 * D[A[x]^2/x, x] + O[x]^(terms+1) // Normal, terms]; CoefficientList[A[x], x] // Rest (* Jean-François Alcover, Apr 06 2012, after Paul D. Hanna, updated Jan 11 2018 *)
a = ConstantArray[0,20]; a[[1]]=1; Do[a[[n]] = (n-1)*Sum[a[[i]]*a[[n-i]],{i,1,n-1}],{n,2,20}]; a (* Vaclav Kotesovec, Feb 22 2014 *)
Module[{max = 20, s}, s = InverseSeries[ComplexExpand[Re[Series[2 DawsonF[x], {x, Infinity, 2 max + 1}]]]]; Table[SeriesCoefficient[s, 2 n - 1] 2^n, {n, 1, max}]] (* Vladimir Reshetnikov, Apr 23 2016 *)

{a(n)=local(A=1+x*O(x^n)); for(i=1, n, A=1+x+x^2*deriv((A-1)^2/x)+x*O(x^n)); polcoeff(A, n)} \\ Paul D. Hanna, Dec 31 2010 [Modified to include a(0) = 1. - Paul D. Hanna, Nov 06 2020]

{a(n) = my(A); A = 1+O(x) ; for( i=0, n, A = 1+x + (A-1)*(2*x*A' - A + 1)); polcoeff(A, n)}; /* Michael Somos, May 12 2012 [Modified to include a(0) = 1. - Paul D. Hanna, Nov 06 2020] */

seq(N) = {
  my(a = vector(N)); a[1] = 1;
  for (n=2, N, a[n] = sum(k=1, n-1, (2*k-1)*a[k]*a[n-k])); a;
};
seq(22)  \\ Gheorghe Coserea, Jan 22 2017

seq(n)={my(g=serlaplace(1 / sqrt(1 - 2*x + O(x*x^n)))); Vec(sqrt((x/serreverse( x*g^2 ))))} \\ Andrew Howroyd, Nov 21 2024

def A000699_list(n):
    list = [1, 1] + [0] * (n - 1)
    for i in range(2, n + 1):
        list[i] = (i - 1) * sum(list[j] * list[i - j] for j in range(1, i))
    return list
print(A000699_list(22)) # M. Eren Kesim, Jun 23 2021

a(n) = (n-1)*Sum_{i=1..n-1} a(i)*a(n-i) for n > 1, with a(1) = a(0) = 1. [Modified to include a(0) = 1. - Paul D. Hanna, Nov 06 2020]
A212273(n) = n * a(n). - Michael Somos, May 12 2012
G.f. satisfies: A(x) = 1 + x + x^2*[d/dx (A(x) - 1)^2/x]. - Paul D. Hanna, Dec 31 2010 [Modified to include a(0) = 1. - Paul D. Hanna, Nov 06 2020]
a(n) ~ n^n * 2^(n+1/2) / exp(n+1) * (1 - 31/(24*n) - 2207/(1152*n^2) - 3085547/(414720*n^3) - 1842851707/(39813120*n^4) - ...). - Vaclav Kotesovec, Feb 22 2014, extended Oct 23 2017
G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 2*x/(A(x) - 3*x/(A(x) - 4*x/(A(x) - 5*x/(A(x) - ...))))), a continued fraction relation. - Paul D. Hanna, Nov 04 2020
G.f. A(x) satisfies: A(x*B(x)^2) = B(x) where B(x) is the g.f. of A001147. - Andrew Howroyd, Nov 21 2024

More terms from David Broadhurst, Dec 14 1999
Inserted "chord" in definition. - N. J. A. Sloane, Jan 19 2017
Added a(0)=1. - N. J. A. Sloane, Nov 05 2020
Modified formulas slightly to include a(0) = 1. - Paul D. Hanna, Nov 06 2020

A099947 Number of topologically connected set partitions of {1,...,n}.

Original entry on oeis.org

1, 1, 1, 1, 2, 6, 21, 85, 385, 1907, 10205, 58455, 355884, 2290536, 15518391, 110283179, 819675482, 6355429550, 51293023347, 430062712439, 3739408304962, 33665192703946, 313354708842791, 3011545611755271, 29847401178719637, 304713973031878687, 3201007359886598431

Offset: 0

N. J. A. Sloane, Nov 12 2004

A set partition of {1,...,n} is topologically connected if the graph whose vertices are the blocks and whose edges are crossing pairs of blocks is connected, where two blocks cross each other if they are of the form {{...x...y...}, {...z...t...}} for some x < z < y < t or z < x < t < y. - Gus Wiseman, Feb 19 2019

			O.g.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 6*x^5 + 21*x^6 + 85*x^7 +...
From _Paul D. Hanna_, Apr 16 2013: (Start)
The o.g.f. satisfies
(1) A(x) = 1 + x/A(x) + 2*x^2/A(x)^2 + 5*x^3/A(x)^3 + 15*x^4/A(x)^4 + 52*x^5/A(x)^5 + 203*x^6/A(x)^6 + ... + A000110(n)*x^n/A(x)^n + ...
(2) A(x) = 1 + x/(A(x)-x) + x^2/((A(x)-x)*(A(x)-2*x)) + x^3/((A(x)-x)*(A(x)-2*x)*(A(x)-3*x)) + x^4/((A(x)-x)*(A(x)-2*x)*(A(x)-3*x)*(A(x)-4*x)) + ... (End)
From _Gus Wiseman_, Feb 19 2019: (Start)
The a(1) = 1 through a(6) = 21 topologically connected set partitions:
  {{1}}  {{12}}  {{123}}  {{1234}}    {{12345}}    {{123456}}
                          {{13}{24}}  {{124}{35}}  {{1235}{46}}
                                      {{13}{245}}  {{124}{356}}
                                      {{134}{25}}  {{1245}{36}}
                                      {{135}{24}}  {{1246}{35}}
                                      {{14}{235}}  {{125}{346}}
                                                   {{13}{2456}}
                                                   {{134}{256}}
                                                   {{1345}{26}}
                                                   {{1346}{25}}
                                                   {{135}{246}}
                                                   {{1356}{24}}
                                                   {{136}{245}}
                                                   {{14}{2356}}
                                                   {{145}{236}}
                                                   {{146}{235}}
                                                   {{15}{2346}}
                                                   {{13}{25}{46}}
                                                   {{14}{25}{36}}
                                                   {{14}{26}{35}}
                                                   {{15}{24}{36}}
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..222
Janet Simpson Beissinger, The enumeration of irreducible combinatorial objects, J. Comb. Theory, Ser. A, 38 (1985), pp. 143-169. (Example 6.2)
Daniel Birmajer, Juan B. Gil, and Michael D. Weiner, A family of Bell transformations, arXiv:1803.07727 [math.CO], 2018.
Kenneth J. Dykema, Generating functions for purely crossing partitions, arXiv:1602.03469 [math.CO], 2016. See column NIS in Table 2 p. 8.
FindStat, The number of topologically connected components of a set partition.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
Martin Klazar, Bell numbers, their relatives and algebraic differential equations
Martin Klazar, Bell numbers, their relatives and algebraic differential equations, J. Combin. Theory, A 102 (2003), 63-87.

Cf. A000108, A000110, A001187, A007297, A016098, A092918, A268814, A268815, A305078, A306438, A323818, A324166, A324172, A324173.

a[0] = 1; a[n_] := Module[{A = 1 + x}, For[i = 1, i <= n, i++, A = Sum[x^m/Product[A - k*x + x*O[x]^n, {k, 1, m}], {m, 0, n}]]; Coefficient[A, x^n]]; Table[a[n], {n, 0, 24}] (* Jean-François Alcover, Sep 13 2013, after Paul D. Hanna *)
nn=8;
nonXQ[stn_]:=!MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Solve[Table[BellB[n]==Sum[Product[a[Length[s]],{s,stn}],{stn,Select[sps[Range[n]],nonXQ]}],{n,nn}],Array[a,nn]] (* Gus Wiseman, Feb 19 2019 *)

{a(n)=if(n<0, 0, polcoeff( x/serreverse(x*serlaplace(exp(exp(x+x*O(x^n))-1))), n))} /* Michael Somos, Sep 22 2005 */

{a(n)=local(A=1+x); for(i=1, n, A=sum(m=0, n, x^m/prod(k=1, m, A - k*x +x*O(x^n)) )); polcoeff(A, n)} \\ Paul D. Hanna, Apr 16 2013

From Paul D. Hanna, Apr 16 2013: (Start)
O.g.f. A(x) satisfies
(1) A(x) = Sum_{n>=0} A000110(n)*x^n/A(x)^n, where A000110 are the Bell numbers.
(2) A(x) = Sum_{n>=0} x^n / Product_{k=1..n} (A(x) - k*x).
(3) A(x) = 1/(1 - x/(A(x) - 1*x/(1 - x/(A(x) - 2*x/(1 - x/(A(x) - 3*x/(1 - x/(A(x) - 4*x/(1 - x/(A(x) - ... )))))))))), a continued fraction. (End)
B(n) = Sum_p Product_{s in p} a(|s|) where p is a non-crossing set partition of {1,...,n} and B = A000110. In words, every set partition of {1,...,n} can be uniquely decomposed as a non-crossing set partition together with a topologically connected set partition of each block. - Gus Wiseman, Feb 19 2019

Name edited by Gus Wiseman, Feb 19 2019

A324170 Numbers whose multiset multisystem (A302242) is crossing.

Original entry on oeis.org

2117, 3973, 4234, 4843, 5183, 5249, 5891, 6351, 6757, 7181, 7801, 7946, 8249, 8468, 8903, 9193, 9686, 9727, 10019, 10063, 10366, 10498, 10585, 11051, 11513, 11567, 11782, 11857, 11919, 12557, 12629, 12702, 12851, 13021, 13193, 13459, 13514, 13631, 14123, 14362

Offset: 1

Gus Wiseman, Feb 17 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset multisystem of n is obtained by taking the multiset of prime indices of each prime index of n.

A multiset of multisets is crossing if it contains a 2-element submultiset of the form {{...x...y...}, {...z...t...}} where x < z < y < t or z < x < t < y.

			The sequence of terms together with their multiset multisystems begins:
  2117: {{1,3},{2,4}}
  3973: {{1,3},{2,5}}
  4234: {{},{1,3},{2,4}}
  4843: {{1,3},{2,6}}
  5183: {{1,1,3},{2,4}}
  5249: {{1,3},{1,2,4}}
  5891: {{1,4},{2,5}}
  6351: {{1},{1,3},{2,4}}
  6757: {{1,3},{2,7}}
  7181: {{1,4},{2,6}}
  7801: {{1,3},{2,8}}
  7946: {{},{1,3},{2,5}}
  8249: {{2,4},{1,2,3}}
  8468: {{},{},{1,3},{2,4}}
  8903: {{1,3},{2,2,4}}
  9193: {{1,3},{1,2,5}}
  9686: {{},{1,3},{2,6}}
  9727: {{1,1,3},{2,5}}

Cf. A000108, A001055, A001263, A002662, A016098, A054726, A056239, A099947, A112798, A302242.

Cf. A324166, A324167, A324168, A324169, A324171, A324173.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
croXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x

A324173 Regular triangle read by rows where T(n,k) is the number of set partitions of {1,...,n} with k topologically connected components.

Original entry on oeis.org

1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 2, 6, 6, 1, 0, 6, 15, 20, 10, 1, 0, 21, 51, 65, 50, 15, 1, 0, 85, 203, 252, 210, 105, 21, 1, 0, 385, 912, 1120, 938, 560, 196, 28, 1, 0, 1907, 4527, 5520, 4620, 2898, 1302, 336, 36, 1, 0, 10205, 24370, 29700, 24780, 15792, 7812, 2730, 540, 45, 1

Offset: 0

Gus Wiseman, Feb 17 2019

A set partition is crossing if it contains a pair of blocks of the form {{...x...y...}, {...z...t...}} where x < z < y < t or z < x < t < y.

The topologically connected components of a set partition correspond to the blocks of its minimal non-crossing coarsening.

			Triangle begins:
     1
     0     1
     0     1     1
     0     1     3     1
     0     2     6     6     1
     0     6    15    20    10     1
     0    21    51    65    50    15     1
     0    85   203   252   210   105    21     1
     0   385   912  1120   938   560   196    28     1
     0  1907  4527  5520  4620  2898  1302   336    36     1
     0 10205 24370 29700 24780 15792  7812  2730   540    45     1
Row n = 4 counts the following set partitions:
  {{1234}}    {{1}{234}}  {{1}{2}{34}}  {{1}{2}{3}{4}}
  {{13}{24}}  {{12}{34}}  {{1}{23}{4}}
              {{123}{4}}  {{12}{3}{4}}
              {{124}{3}}  {{1}{24}{3}}
              {{134}{2}}  {{13}{2}{4}}
              {{14}{23}}  {{14}{2}{3}}

FindStat, The number of topologically connected components of a set partition.

Row sums are A000110. Column k = 1 is A099947.
Cf. A000108, A001263, A002061, A002662, A007297, A016098, A048993, A054726, A293510, A305078, A305079, A323818.
Cf. A324166, A324167, A324169, A324170, A324171, A324172.

croXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
crosscmpts[stn_]:=csm[Union[Subsets[stn,{1}],Select[Subsets[stn,{2}],croXQ]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
Table[Length[Select[sps[Range[n]],Length[crosscmpts[#]]==k&]],{n,0,8},{k,0,n}]

A136653 G.f.: A(x) satisfies: coefficient of x^n in A(x)^(n+1)/(n+1) = 2^(n*(n-1)/2).

Original entry on oeis.org

1, 1, 1, 4, 39, 748, 27162, 1880872, 252273611, 66358216668, 34506398937158, 35644762692112792, 73356520492898454022, 301274559225693420690360, 2471654510727312089903896948, 40527708183358718551543295827536, 1328579216048284168977214446788083699

Offset: 0

Paul D. Hanna, Jan 15 2008

nonn

a(n) is the number of graphs on vertices 1,...,n such that, when these vertices are arranged counterclockwise around a circle and edges are drawn as straight line segments, the resulting diagram is connected. - Jonathan Novak (j2novak(AT)math.uwaterloo.ca), Apr 30 2010

In this interpretation, both intersecting (set theoretically) and crossing (topologically) edges are considered connected. - Gus Wiseman, Feb 23 2019

			G.f.: A(x) = 1 + x + x^2 + 4*x^3 + 39*x^4 + 748*x^5 + 27162*x^6 +...
Let F(x) = 1 + x + 2*x^2 + 8*x^3 + 64*x^4 + 1024*x^5 +...+ 2^(n*(n-1)/2)*x^n +..
then A(x) = F(x/A(x)), A(x*F(x)) = F(x).
Coefficient of x^n in A(x)^(n+1)/(n+1) = 2^(n*(n-1)/2),
as can be seen by the main diagonal in the array of
coefficients in the initial powers of A(x):
A^1: [(1), 1, 1, 4, 39, 748, 27162, 1880872, 252273611,...;
A^2: [1, (2), 3, 10, 87, 1582, 55914, 3817876, 508370795,...;
A^3: [1, 3, (6), 19, 147, 2517, 86398, 5813550, 768378627,...;
A^4: [1, 4, 10, (32), 223, 3572, 118778, 7870640, 1032387787,...;
A^5: [1, 5, 15, 50, (320), 4771, 153245, 9992130, 1300492845,...;
A^6: [1, 6, 21, 74, 444, (6144), 190023, 12181278, 1572792585,...;
A^7: [1, 7, 28, 105, 602, 7728, (229376), 14441659, 1849390375,...;
A^8: [1, 8, 36, 144, 802, 9568, 271616, (16777216), 2130394591,...;
A^9: [1, 9, 45, 192, 1053, 11718, 317112, 19192320, (2415919104),...;
dividing each diagonal term in row n by (n+1) gives 2^(n*(n-1)/2).
The diagonal above the main diagonal gives coefficients of l.g.f.:
log(F(x)) = x + 3*x^2/2 + 19*x^3/3 + 223*x^4/4 + 4771*x^5/5 +...

Andrew Howroyd, Table of n, a(n) for n = 0..50
Gus Wiseman, The a(4) = 39 graphs connected by overlaps or crossings.

Cf. A136652 (log(A(x))), A136654.
Cf. A000699, A002662, A006125, A007297, A016098, A054726, A099947, A306438.
Cf. A324166, A324169, A324172, A324173, A324327, A324328.

max = 15; s = x*Sum[2^(k*(k-1)/2)*x^k, {k, 0, max}] + O[x]^(max+2); x/InverseSeries[s] + O[x]^(max+1) // CoefficientList[#, x]& (* Jean-François Alcover, Sep 03 2017 *)
croXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x0]&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
bicmpts[stn_]:=csm[Union[Subsets[stn,{1}],Select[Subsets[stn,{2}],Intersection@@#!={}&],Select[Subsets[stn,{2}],croXQ]]];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],And[Union@@#==Range[n],Length[bicmpts[#]]<=1]&]],{n,0,5}] (* Gus Wiseman, Feb 23 2019 *)

a(n)=polcoeff(x/serreverse(x*sum(k=0,n,2^(k*(k-1)/2)*x^k +x*O(x^n))),n)

G.f.: A(x) = x/Series_Reversion( x*Sum_{k=0..n} 2^(k(k-1)/2)*x^k ).

Equals the free cumulant sequence corresponding to A006125. - Jonathan Novak (j2novak(AT)math.uwaterloo.ca), Apr 30 2010

Name changed and part of prior name moved to formula section by Paul D. Hanna, Sep 19 2013

A324172 Number of subsets of {1,...,n} that cross their complement.

Original entry on oeis.org

0, 0, 0, 0, 2, 10, 32, 84, 198, 438, 932, 1936, 3962, 8034, 16200, 32556, 65294, 130798, 261836, 523944, 1048194, 2096730, 4193840, 8388100, 16776662, 33553830, 67108212, 134217024, 268434698, 536870098, 1073740952, 2147482716, 4294966302, 8589933534, 17179868060

Offset: 0

Gus Wiseman, Feb 17 2019

Two sets cross each other if they are of the form {{...x...y...}, {...z...t...}} where x < z < y < t or z < x < t < y.

Also the number of verex cuts in the wheel graph on n nodes. - Eric W. Weisstein, Apr 22 2023

			The a(5) = 10 subsets are {1,3}, {1,4}, {2,4}, {2,5}, {3,5}, {1,2,4}, {1,3,4}, {1,3,5}, {2,3,5}, {2,4,5}.

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-9,7,-2).
Eric Weisstein's World of Mathematics, Vertex Cut
Eric Weisstein's World of Mathematics, Wheel Graph

Cf. A000108, A000110, A000124, A001263, A002061, A002662, A016098, A306438.

Cf. A324166, A324167, A324168, A324169, A324173.

croXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x

concat([0,0,0,0], Vec(2*x^4 / ((1 - x)^3*(1 - 2*x)) + O(x^40))) \\ Colin Barker, Feb 19 2019

a(0) = 0; a(n) = 2^n - n^2 + n - 2.
a(n) = 2*A002662(n-1) for n > 0.
G.f.: 2*x^4/((1-2*x)*(1-x)^3).
a(n) = 5*a(n-1) - 9*a(n-2) + 7*a(n-3) - 2*a(n-4) for n>4. - Colin Barker, Feb 18 2019

A324167 Number of non-crossing antichain covers of {1,...,n}.

Original entry on oeis.org

1, 1, 2, 9, 67, 633, 6763, 77766, 938957, 11739033, 150649945, 1973059212, 26265513030, 354344889798, 4833929879517, 66568517557803, 924166526830701, 12920482325488761, 181750521972603049, 2570566932237176232, 36532394627404815308, 521439507533582646156

Offset: 0

Gus Wiseman, Feb 17 2019

nonn

An antichain is non-crossing if no pair of distinct parts is of the form {{...x...y...}, {...z...t...}} where x < z < y < t or z < x < t < y.

			The a(3) = 9 antichains:
  {{1,2,3}}
  {{1},{2,3}}
  {{2},{1,3}}
  {{3},{1,2}}
  {{1,2},{1,3}}
  {{1,2},{2,3}}
  {{1,3},{2,3}}
  {{1},{2},{3}}
  {{1,2},{1,3},{2,3}}

Andrew Howroyd, Table of n, a(n) for n = 0..500

Cf. A000108, A000124, A000372 (antichains), A001006, A006126 (antichain covers), A014466, A048143, A054726 (non-crossing graphs), A099947, A261005, A283877, A306438.

Cf. A324166, A324168, A324169, A324170, A324171, A324173, A359984 (no singletons).

nn=6;
croXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x

seq(n)={my(f=O(1)); for(n=2, n, f = 1 + (4*x + x^2)*f^2 - 3*x^2*(1 + x)*f^3); Vec(subst(x*(1 + x^2*f^2 - 3*x^3*f^3), x, x/(1-x))/x) } \\ Andrew Howroyd, Jan 20 2023

Inverse binomial transform of A324168.

Binomial transform of A359984. - Andrew Howroyd, Jan 20 2023

Terms a(9) and beyond from Andrew Howroyd, Jan 20 2023

A324171 Number of non-crossing multiset partitions of normal multisets of size n.

Original entry on oeis.org

1, 1, 4, 16, 75, 378, 2042, 11489, 66697

Offset: 0

Gus Wiseman, Feb 17 2019

A multiset is normal if its union is an initial interval of positive integers.

A multiset partition is crossing if it has a 2-element submultiset of the form {{...x...y...}, {...z...t...}} where x < z < y < t or z < x < t < y.

			The A255906(5) - a(5) = 22 crossing multiset partitions:
  {{13}{124}}  {{1}{13}{24}}
  {{13}{224}}  {{1}{24}{35}}
  {{13}{234}}  {{2}{13}{24}}
  {{13}{244}}  {{2}{14}{35}}
  {{13}{245}}  {{3}{13}{24}}
  {{14}{235}}  {{3}{14}{25}}
  {{24}{113}}  {{4}{13}{24}}
  {{24}{123}}  {{4}{13}{25}}
  {{24}{133}}  {{5}{13}{24}}
  {{24}{134}}
  {{24}{135}}
  {{25}{134}}
  {{35}{124}}

Cf. A000108 (non-crossing set partitions), A000124, A001006, A001055, A001263, A007297, A054726 (non-crossing graphs), A099947, A194560, A255906 (multiset partitions of normal multisets), A306438.

Cf. A324166, A324167, A324168, A324169, A324170, A324173.

nonXQ[stn_]:=!MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
Table[Sum[Length[Select[mps[m],nonXQ]],{m,allnorm[n]}],{n,0,8}]

A324168 Number of non-crossing antichains of nonempty subsets of {1,...,n}.

Original entry on oeis.org

1, 2, 5, 19, 120, 1084, 11783, 141110, 1791156, 23646352, 321220257, 4459886776, 63000867229, 902528825332, 13080523942476, 191445447535373, 2825542818304080, 42005234042942228, 628422035415996065, 9454076958795999908, 142933849346150225253, 2170556938059142024688

Offset: 0

Gus Wiseman, Feb 17 2019

nonn

An antichain is non-crossing if no pair of distinct parts is of the form {{...x...y...}, {...z...t...}} where x < z < y < t or z < x < t < y.

			The a(0) = 1 through a(3) = 19 non-crossing antichains:
  {}  {}     {}        {}
      {{1}}  {{1}}     {{1}}
             {{2}}     {{2}}
             {{12}}    {{3}}
             {{1}{2}}  {{12}}
                       {{13}}
                       {{23}}
                       {{123}}
                       {{1}{2}}
                       {{1}{3}}
                       {{2}{3}}
                       {{1}{23}}
                       {{2}{13}}
                       {{3}{12}}
                       {{12}{13}}
                       {{12}{23}}
                       {{13}{23}}
                       {{1}{2}{3}}
                       {{12}{13}{23}}

Andrew Howroyd, Table of n, a(n) for n = 0..500

Cf. A000108 (non-crossing set partitions), A000124, A000372 (antichains), A001006, A001263, A006126 (antichain covers), A014466 (nonempty antichains), A054726 (non-crossing graphs), A099947, A261005, A306438.

Cf. A324166, A324167, A324169, A324170, A324171, A324173, A359984.

nn=6;
nonXQ[stn_]:=!MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x

seq(n)={my(f=O(1)); for(n=2, n, f = 1 + (4*x + x^2)*f^2 - 3*x^2*(1 + x)*f^3); Vec(subst(x*(1 + x^2*f^2 - 3*x^3*f^3), x, x/(1-2*x))/x) } \\ Andrew Howroyd, Jan 20 2023

Binomial transform of A324167.

G.f.: A(x) = B(x/(1-2*x))/x where B(x)/x is the g.f. of A359984. - Andrew Howroyd, Jan 20 2023

Terms a(9) and beyond from Andrew Howroyd, Jan 20 2023

A324324 MM-numbers of crossing set partitions.

Original entry on oeis.org

2117, 3973, 4843, 5891, 6757, 7181, 7801, 10019, 10063, 11051, 11567, 13021, 13193, 13459, 14123, 14921, 17603, 18407, 18761, 18877, 19307, 19633, 20941, 21083, 21251, 21457, 22849, 23519, 23533, 24727, 26101, 27133, 27169, 27173, 27413, 29111, 30479, 31261

Offset: 1

Gus Wiseman, Feb 22 2019

nonn

A multiset multisystem is a finite multiset of finite multisets. A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. The multiset multisystem with MM-number n is formed by taking the multiset of prime indices of each part in the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset multisystem with MM-number 78 is {{},{1},{1,2}}.

A multiset multisystem is crossing if it contains two parts of the form {{...x...y...},{...z...t...}} with x < z < y < t or z < x < t < y.

Cf. A000108 (non-crossing set partitions), A001055, A001222, A003963, A005117, A016098 (crossing set partitions), A054726, A056239, A112798, A302242, A302243, A302505, A302521 (MM-numbers of set partitions).

Cf. A324166, A324167, A324169, A324170, A324171, A324326.

croXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;xTable[PrimePi[p],{k}]]]];
setptnQ[bks_]:=UnsameQ@@Join@@bks&&!MemberQ[bks,{}];
Select[Range[10000],And[croXQ[primeMS/@primeMS[#]],setptnQ[primeMS/@primeMS[#]]]&]

cp's OEIS Frontend

A000699 Number of irreducible chord diagrams with 2n nodes.

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A099947 Number of topologically connected set partitions of {1,...,n}.

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A324170 Numbers whose multiset multisystem (A302242) is crossing.

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A324173 Regular triangle read by rows where T(n,k) is the number of set partitions of {1,...,n} with k topologically connected components.

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A136653 G.f.: A(x) satisfies: coefficient of x^n in A(x)^(n+1)/(n+1) = 2^(n*(n-1)/2).

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A324172 Number of subsets of {1,...,n} that cross their complement.

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A324167 Number of non-crossing antichain covers of {1,...,n}.

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