A326211 -id:A326211 - cp's OEIS frontend

A326246 Number of crossing, capturing set partitions of {1..n}.

0, 0, 0, 0, 0, 3, 37, 307, 2173, 14344, 92402, 596688

Offset: 0

Gus Wiseman, Jun 20 2019

A set 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, and capturing if it has two blocks of the form {...x...y...}, {...z...t...} where x < z < t < y or z < x < y < t. Capturing is a weaker condition than nesting, so for example {{1,3,5},{2,4}} is capturing but not nesting.

			The a(5) = 3 set partitions:
  {{1,3,4},{2,5}}
  {{1,3,5},{2,4}}
  {{1,4},{2,3,5}}

Eric Marberg, Crossings and nestings in colored set partitions, arXiv preprint arXiv:1203.5738 [math.CO], 2012.

MM-numbers of crossing, capturing multiset partitions are A326259.
Crossing set partitions are A016098.
Capturing set partitions are A326243.
Crossing, nesting set partitions are A326248.
Crossing, non-capturing set partitions are A326245.
Non-crossing, capturing set partitions are A122880 (conjecture).
Cf. A000108, A000110, A058681, A099947, A324170, A326211, A326249, A054391, A326255.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
croXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x

A326259 MM-numbers of crossing, capturing multiset partitions (with empty parts allowed).

Original entry on oeis.org

8903, 15167, 16717, 17806, 18647, 20329, 20453, 21797, 22489, 25607, 26709, 27649, 29551, 30334, 31373, 32741, 33434, 34691, 35177, 35612, 35821, 37091, 37133, 37294, 37969, 38243, 39493, 40658, 40906, 41449, 42011, 42949, 43594, 43817, 43873, 44515, 44861

Offset: 1

Gus Wiseman, Jun 22 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 with MM-number n is obtained by taking the multiset of prime indices of each prime index of n.

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. It is capturing if it has two blocks of the form {...x...y...} and {...z...t...} where x < z and y > t or x > z and y < t. Capturing is a weaker condition than nesting, so for example {{1,3,5},{2,4}} is capturing but not nesting.

			The sequence of terms together with their multiset multisystems begins:
   8903: {{1,3},{2,2,4}}
  15167: {{1,3},{2,2,5}}
  16717: {{2,4},{1,3,3}}
  17806: {{},{1,3},{2,2,4}}
  18647: {{1,3},{2,2,6}}
  20329: {{1,3},{1,2,2,4}}
  20453: {{1,2,3},{1,2,4}}
  21797: {{1,1,3},{2,2,4}}
  22489: {{1,4},{2,2,5}}
  25607: {{1,3},{2,2,7}}
  26709: {{1},{1,3},{2,2,4}}
  27649: {{1,4},{2,2,6}}
  29551: {{1,3},{2,2,8}}
  30334: {{},{1,3},{2,2,5}}
  31373: {{2,5},{1,3,3}}
  32741: {{1,3},{2,2,2,4}}
  33434: {{},{2,4},{1,3,3}}
  34691: {{1,2,3},{2,2,4}}
  35177: {{1,3},{1,2,2,5}}
  35612: {{},{},{1,3},{2,2,4}}

Crossing set partitions are A000108.
Capturing set partitions are A326243.
Crossing, capturing set partitions are A326246.
MM-numbers of crossing multiset partitions are A324170.
MM-numbers of nesting multiset partitions are A326256.
MM-numbers of capturing multiset partitions are A326255.
MM-numbers of unsortable multiset partitions are A326258.
Cf. A001055, A001519, A016098, A056239, A058681, A112798, A122880, A302242.
Cf. A326211, A326245, A326248, A326249, A054391.

croXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;xTable[PrimePi[p],{k}]]]];
Select[Range[100000],capXQ[primeMS/@primeMS[#]]&&croXQ[primeMS/@primeMS[#]]&]

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).

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).

A326334 Number of sortable factorizations of n.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 7, 2, 2, 3, 4, 1, 4, 1, 7, 2, 2, 2, 8, 1, 2, 2, 7, 1, 4, 1, 4, 4, 2, 1, 12, 2, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 8, 1, 2, 4, 11, 2, 4, 1, 4, 2, 4, 1, 14, 1, 2, 4, 4, 2, 4, 1, 12, 5, 2, 1, 8, 2, 2

Offset: 1

Gus Wiseman, Jun 27 2019

nonn

A factorization into factors > 1 is sortable if there is a permutation (c_1,...,c_k) of the factors such that the maximum prime factor (in the standard factorization of an integer into prime numbers) 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) = 16 sortable factorizations:
  (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)
Missing from this list are the following 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)

Factorizations are A001055.
Unsortable factorizations are A326291.
Sortable integer partitions are A326333.
Cf. A058681, A326211, A326212, A326237, A326258, A326332.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],OrderedQ[Join@@Sort[First/@FactorInteger[#]&/@#,OrderedQ[PadRight[{#1,#2}]]&]]&]],{n,100}]

A001055(n) = a(n) + A326291(n).

A326277 Number of crossing normal multiset partitions of weight n.

Original entry on oeis.org

0, 0, 0, 0, 1, 22, 314, 3711, 39947

Offset: 0

Gus Wiseman, Jun 22 2019

A multiset partition is normal if it covers an initial interval of positive integers.

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.

			The a(5) = 22 crossing normal multiset partitions:
  {{1,3},{1,2,4}}  {{1},{1,3},{2,4}}
  {{1,3},{2,2,4}}  {{1},{2,4},{3,5}}
  {{1,3},{2,3,4}}  {{2},{1,3},{2,4}}
  {{1,3},{2,4,4}}  {{2},{1,4},{3,5}}
  {{1,3},{2,4,5}}  {{3},{1,3},{2,4}}
  {{1,4},{2,3,5}}  {{3},{1,4},{2,5}}
  {{2,4},{1,1,3}}  {{4},{1,3},{2,4}}
  {{2,4},{1,2,3}}  {{4},{1,3},{2,5}}
  {{2,4},{1,3,3}}  {{5},{1,3},{2,4}}
  {{2,4},{1,3,4}}
  {{2,4},{1,3,5}}
  {{2,5},{1,3,4}}
  {{3,5},{1,2,4}}

Crossing simple graphs are A326210.
Normal multiset partitions are A255906.
Non-crossing normal multiset partitions are A324171.
MM-numbers of crossing multiset partitions are A324170.
Cf. A000108, A016098, A054726, A058681.
Cf. A326211, A326212, A326255, A326256, A326258.

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

cp's OEIS Frontend

A326246 Number of crossing, capturing set partitions of {1..n}.

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A326259 MM-numbers of crossing, capturing multiset partitions (with empty parts allowed).

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

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

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A326291 Number of unsortable factorizations of n.

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

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A326334 Number of sortable factorizations of n.

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A326277 Number of crossing normal multiset partitions of weight n.

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