A326209 -id:A326209 - cp's OEIS frontend

A326252 Number of digraphs with vertices {1..n} whose increasing edges are crossing.

0, 0, 0, 0, 16384, 22020096, 62679678976, 556181084962816

Offset: 0

Gus Wiseman, Jun 30 2019

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.

Simple graphs whose edges are crossing are A326210.
Digraphs whose increasing edges are not crossing are A326251.
Digraphs whose edges are not crossing are A326237.
Cf. A002416, A002720, A229865, A324170, A326209, A326252, A326258.

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

a(n) = 2^(n * (n + 1)/2) * A326210(n).

A326279 Number of labeled n-vertex simple graphs containing either a crossing or a nesting pair of edges.

Original entry on oeis.org

0, 0, 0, 0, 28, 864, 32064, 2094064

Offset: 0

Gus Wiseman, Jun 23 2019

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

			The a(4) = 28 edge-sets:
  {13,24}  {12,13,24}  {12,13,14,23}  {12,13,14,23,24}  {12,13,14,23,24,34}
  {14,23}  {12,14,23}  {12,13,14,24}  {12,13,14,23,34}
           {13,14,23}  {12,13,23,24}  {12,13,14,24,34}
           {13,14,24}  {12,13,24,34}  {12,13,23,24,34}
           {13,23,24}  {12,14,23,24}  {12,14,23,24,34}
           {13,24,34}  {12,14,23,34}  {13,14,23,24,34}
           {14,23,24}  {13,14,23,24}
           {14,23,34}  {13,14,23,34}
                       {13,14,24,34}
                       {13,23,24,34}
                       {14,23,24,34}

Crossing and nesting simple graphs are (both) A326210, while non-crossing, non-nesting simple graphs are A326244.
Cf. A000108, A001519, A006125, A016098, A054726, A095661, A324170.
Cf. A326209, A326211, A326248, A326250, A326256.

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

A006125(n) = a(n) + A326244(n).

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).

A326332 Number of integer partitions of n with unsortable prime factors.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 5, 9, 14, 22, 33, 50, 71, 100, 140, 196, 265, 360, 480, 641, 842, 1104, 1432, 1855, 2378, 3040, 3858, 4888, 6146, 7708, 9616, 11969, 14818, 18305, 22511, 27629, 33773, 41191, 50069, 60744, 73453, 88645, 106681

Offset: 0

Gus Wiseman, Jun 27 2019

nonn

An integer partition has unsortable prime factors if there is no permutation (c_1,...,c_k) of the parts such that the maximum prime factor of c_i is at most the minimum prime factor of c_{i+1}. For example, the partition (27,8,6) is sortable because the permutation (8,6,27) satisfies the condition.

			The a(12) = 1 through a(17) = 14 partitions:
  (6,6)  (10,3)   (6,6,2)    (6,6,3)      (10,6)         (14,3)
         (6,6,1)  (10,3,1)   (10,3,2)     (6,6,4)        (6,6,5)
                  (6,6,1,1)  (6,6,2,1)    (10,3,3)       (10,4,3)
                             (10,3,1,1)   (6,6,2,2)      (10,6,1)
                             (6,6,1,1,1)  (6,6,3,1)      (6,6,3,2)
                                          (10,3,2,1)     (6,6,4,1)
                                          (6,6,2,1,1)    (10,3,2,2)
                                          (10,3,1,1,1)   (10,3,3,1)
                                          (6,6,1,1,1,1)  (6,6,2,2,1)
                                                         (6,6,3,1,1)
                                                         (10,3,2,1,1)
                                                         (6,6,2,1,1,1)
                                                         (10,3,1,1,1,1)
                                                         (6,6,1,1,1,1,1)

Sortable integer partitions are A326333.
Unsortable set partitions are A058681.
Unsortable normal multiset partitions are A326211.
MM-numbers of unsortable multiset partitions are A326258.
Cf. A000041, A000108, A001055, A056239, A112798, A326209, A326212.

Table[Length[Select[IntegerPartitions[n],!OrderedQ[Join@@Sort[First/@FactorInteger[#]&/@#,OrderedQ[PadRight[{#1,#2}]]&]]&]],{n,0,20}]

A000041(n) = a(n) + A326333(n).

A326291 Number of unsortable factorizations 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, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 3, 0, 0, 0

Offset: 1

Gus Wiseman, Jun 24 2019

nonn

A factorization into factors > 1 is unsortable if there is no permutation (c_1,...,c_k) of the factors such that the maximum prime factor of c_i is at most the minimum prime factor of c_{i+1}. For example, the factorization (6*8*27) is sortable because the permutation (8,6,27) satisfies the condition.

			The a(180) = 10 unsortable factorizations:
  (2*3*3*10)  (5*6*6)   (3*60)
              (2*3*30)  (6*30)
              (2*9*10)  (9*20)
              (3*3*20)  (10*18)
              (3*6*10)
Missing from this list are:
  (2*2*3*3*5)  (2*2*5*9)   (4*5*9)   (2*90)   (180)
               (2*3*5*6)   (2*2*45)  (4*45)
               (3*3*4*5)   (2*5*18)  (5*36)
               (2*2*3*15)  (2*6*15)  (12*15)
                           (3*4*15)
                           (3*5*12)

Unsortable set partitions are A058681.
Unsortable normal multiset partitions are A326211.
MM-numbers of unsortable multiset partitions are A326258.
Cf. A000108, A001055, A001519, A016098, A112798, A302242, A324170, A324171.
Cf. A326209, A326212, A326243, A326256, A326257.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
lexsort[f_,c_]:=OrderedQ[PadRight[{f,c}]];
primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Select[facs[n],!OrderedQ[Join@@Sort[primeMS/@#,lexsort]]&]],{n,100}]

A326333 Number of integer partitions of n with sortable prime factors.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, 56, 76, 99, 132, 171, 222, 283, 363, 457, 577, 721, 902, 1115, 1379, 1693, 2076, 2530, 3077, 3723, 4500, 5410, 6494, 7765, 9270, 11025, 13089, 15491, 18307, 21569, 25369, 29765, 34869, 40750, 47546, 55361, 64367, 74685, 86529

Offset: 0

Gus Wiseman, Jun 27 2019

nonn

An integer partition has sortable prime factors if there is a permutation (c_1,...,c_k) of the parts such that the maximum prime factor of c_i is at most the minimum prime factor of c_{i+1}. For example, the partition (27,8,6) is sortable because the permutation (8,6,27) satisfies the condition.

Unsortable integer partitions are A326332.
Sortable normal multiset partitions are A326212.
Sortable factorizations are A326334.
Cf. A000041, A000108, A001055, A058681, A112798, A326209, A326211, A326258.

Table[Length[Select[IntegerPartitions[n],OrderedQ[Join@@Sort[First/@FactorInteger[#]&/@#,OrderedQ[PadRight[{#1,#2}]]&]]&]],{n,0,20}]

A000041(n) = a(n) + A326332(n).

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).

cp's OEIS Frontend

A326252 Number of digraphs with vertices {1..n} whose increasing edges are crossing.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A326279 Number of labeled n-vertex simple graphs containing either a crossing or a nesting pair of edges.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

A326332 Number of integer partitions of n with unsortable prime factors.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A326291 Number of unsortable factorizations of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A326333 Number of integer partitions of n with sortable prime factors.

Views

Author

Keywords

Comments

Crossrefs

Programs

Mathematica

Formula

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

Formula