A326252 -id:A326252 - cp's OEIS frontend

A229865 Number of n X n 0..1 arrays with corresponding row and column sums equal.

1, 2, 8, 80, 2432, 247552, 88060928, 112371410944, 523858015518720, 9041009511609073664, 583447777113052431515648, 141885584718620229407228821504, 130832005909904417592540055577034752, 459749137931232137234615429529864283095040, 6182706200522446492946534924719926752508110700544

Offset: 0

R. H. Hardin, Oct 01 2013

nonn

Also known as labeled Eulerian digraphs allowing loops. - Brendan McKay, May 12 2019

			Some solutions for n=4:
  0 0 0 1     0 0 1 0     0 0 0 1     0 0 1 0     0 0 1 1
  0 1 0 0     1 0 0 0     1 0 1 0     0 0 1 1     1 0 0 1
  0 0 0 1     0 1 0 0     0 1 0 1     0 1 1 1     1 1 1 0
  1 0 1 0     0 0 0 1     0 1 1 0     1 1 0 0     0 1 1 1
From _Gus Wiseman_, Jun 22 2019: (Start)
The a(3) = 8 Eulerian digraph edge-sets:
  {}
  {11}
  {22}
  {11,22}
  {12,21}
  {11,12,21}
  {12,21,22}
  {11,12,21,22}
(End)

Mohammad Behzad Kang and Andrew Salch, The mod p cohomology of the Morava stabilizer group at large primes, arXiv:2410.24171 [math.AT], 2024. See p. 46.

Column 1 of A229870.
The unlabeled version is A308111.
Cf. A000595, A002416, A002720, A007080.
Cf. A326209, A326237, A326251, A326252, A326253.

Table[Length[Select[Subsets[Tuples[Range[n],2]],Sort[First/@#]==Sort[Last/@#]&]],{n,4}] (* Gus Wiseman, Jun 22 2019 *)

a(n) = 2^n * A007080(n). - Andrew Howroyd, Sep 11 2019

a(0)=1 prepended by Alois P. Heinz, May 14 2019

Terms a(11) and beyond from Andrew Howroyd, Sep 11 2019

A326251 Number of digraphs with vertices {1..n} whose increasing edges are not crossing.

Original entry on oeis.org

1, 2, 16, 512, 49152, 11534336, 6039797760, 6768868458496, 15885743998107648, 77083611222073409536, 767126299049285413502976, 15572324598183490228037091328, 642316330843573124053884695740416, 53681919993405760099480940765478125568

Offset: 0

Gus Wiseman, Jun 30 2019

nonn

A directed edge (a,b) is increasing if a < b. Two edges (a,b), (c,d) are crossing if a < c < b < d or c < a < d < b.

Conjecture: Also the number of non-nesting digraphs with vertices {1..n} whose increasing edges are not crossing, where two edges (a,b), (c,d) are nesting if a < c < d < b or c < a < b < d.

Simple graphs whose edges are non-crossing are A054726.
Digraphs whose edges are not crossing are A326237.
Digraphs whose increasing edges are crossing are A326252.
Cf. A002416, A229865, A324170, A326209, A326250, A326251.

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

a(n) = 2^(n * (n + 1)/2) * A054726(n).

A326290 Number of non-crossing n-vertex graphs with loops.

Original entry on oeis.org

1, 2, 8, 64, 768, 11264, 184320, 3227648, 59179008, 1121714176, 21803040768, 432218832896, 8705009516544, 177618573852672, 3663840373899264, 76277945940836352, 1600706475536154624, 33823752545680490496, 719051629204296695808, 15368152475218787434496

Offset: 0

Gus Wiseman, Sep 12 2019

nonn

Two edges {a,b}, {c,d} are crossing if a < c < b < d or c < a < d < b.

			The a(0) = 1 through a(2) = 8 non-crossing edge sets with loops:
  {}  {}    {}
      {11}  {11}
            {12}
            {22}
            {11,12}
            {11,22}
            {12,22}
            {11,12,22}

Andrew Howroyd, Table of n, a(n) for n = 0..200

Crossing and nesting simple graphs are (both) A326210, while non-crossing, non-nesting simple graphs are A326244.

Cf. A000108, A002061, A002662, A054726, A324172, A326252.

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

seq(n)=Vec(1+3*x-4*x^2 -x*sqrt(1-24*x+16*x^2 + O(x^n))) \\ Andrew Howroyd, Sep 14 2019

From Andrew Howroyd, Sep 14 2019: (Start)
a(n) = 2^n * A054726(n).
G.f.: 1 + 3*x - 4*x^2 - x*sqrt(1 - 24*x + 16*x^2). (End)

Terms a(6) and beyond from Andrew Howroyd, Sep 14 2019

A326253 Number of sequences of distinct ordered pairs of positive integers up to n.

Original entry on oeis.org

1, 2, 65, 986410, 56874039553217, 42163840398198058854693626, 1011182700521015817607065606491025592595137, 1653481537585545171449931620186035466059689728986775126016505970

Offset: 0

Gus Wiseman, Jun 21 2019

nonn

			The a(2) = 65 sequences:
  ()  (11)  (11,12)  (11,12,21)  (11,12,21,22)
      (12)  (11,21)  (11,12,22)  (11,12,22,21)
      (21)  (11,22)  (11,21,12)  (11,21,12,22)
      (22)  (12,11)  (11,21,22)  (11,21,22,12)
            (12,21)  (11,22,12)  (11,22,12,21)
            (12,22)  (11,22,21)  (11,22,21,12)
            (21,11)  (12,11,21)  (12,11,21,22)
            (21,12)  (12,11,22)  (12,11,22,21)
            (21,22)  (12,21,11)  (12,21,11,22)
            (22,11)  (12,21,22)  (12,21,22,11)
            (22,12)  (12,22,11)  (12,22,11,21)
            (22,21)  (12,22,21)  (12,22,21,11)
                     (21,11,12)  (21,11,12,22)
                     (21,11,22)  (21,11,22,12)
                     (21,12,11)  (21,12,11,22)
                     (21,12,22)  (21,12,22,11)
                     (21,22,11)  (21,22,11,12)
                     (21,22,12)  (21,22,12,11)
                     (22,11,12)  (22,11,12,21)
                     (22,11,21)  (22,11,21,12)
                     (22,12,11)  (22,12,11,21)
                     (22,12,21)  (22,12,21,11)
                     (22,21,11)  (22,21,11,12)
                     (22,21,12)  (22,21,12,11)

Cf. A000522, A000595, A002416, A095661, A229865, A326209, A326237, A326251, A326252, A326258.

Table[Sum[k!*Binomial[n^2,k],{k,0,n^2}],{n,0,4}]

a(n) = A000522(n^2).

A326292 Number of crossing integer partitions of n.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 43, 57, 80, 105, 142, 186, 248, 320, 421, 539, 698, 889, 1140, 1438, 1827, 2291, 2882, 3593, 4489, 5559, 6902, 8503, 10484, 12853, 15763

Offset: 0

Gus Wiseman, Oct 03 2019

nonn

A multiset partition is crossing if it has two blocks of the form {...x...y...}, {...z...t...} where x < z < y < t or z < x < t < y. An integer partition is crossing if, by replacing each part with its multiset of prime indices, we obtain a crossing multiset partition.

			The a(31) = 1 through a(36) = 7 partitions:
  21,10  21,10,1  21,10,2    21,10,3      21,10,4        21,10,5
                  21,10,1,1  21,10,2,1    21,10,2,2      21,10,3,2
                             21,10,1,1,1  21,10,3,1      21,10,4,1
                                          21,10,2,1,1    21,10,2,2,1
                                          21,10,1,1,1,1  21,10,3,1,1
                                                         21,10,2,1,1,1
                                                         21,10,1,1,1,1,1

The Heinz numbers of these partitions are given by A324170.

Cf. A000108, A002662, A016098, A056239, A112798, A324172, A324326, A326210, A326252, A326332.

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

More terms from Jinyuan Wang, Jun 28 2020

cp's OEIS Frontend

A229865 Number of n X n 0..1 arrays with corresponding row and column sums equal.

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A326251 Number of digraphs with vertices {1..n} whose increasing edges are not crossing.

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A326290 Number of non-crossing n-vertex graphs with loops.

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A326253 Number of sequences of distinct ordered pairs of positive integers up to n.

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A326292 Number of crossing integer partitions of n.

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