A324171 -id:A324171 - cp's OEIS frontend

A324326 Number of crossing multiset partitions of a multiset whose multiplicities are the prime indices of n.

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 10, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 31, 0, 0, 0, 0, 0, 36, 0, 14, 0, 0, 0, 25, 0, 0, 0, 71, 0, 0, 0, 0, 0, 0, 0, 103, 0, 0, 0, 0, 0, 0, 0, 75

Offset: 1

Gus Wiseman, Feb 22 2019

nonn

This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

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

			The a(36) = 10 crossing multiset partitions of {1,1,2,2,3,4}:
  {{1,3},{1,2,2,4}}
  {{2,4},{1,1,2,3}}
  {{1,1,3},{2,2,4}}
  {{1,2,3},{1,2,4}}
  {{1},{1,3},{2,2,4}}
  {{1},{2,4},{1,2,3}}
  {{2},{1,3},{1,2,4}}
  {{2},{1,1,3},{2,4}}
  {{1,2},{1,3},{2,4}}
  {{1},{2},{1,3},{2,4}}

Cf. A000108, A001055, A001970, A016098, A054726, A099947, A181821, A305936, A306438, A318284, A318285.

Cf. A324167, A324168, A324169, A324170, A324171, A324324, A324325.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
croXQ[stn_]:=MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x

a(n) + A324325(n) = A318284(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}]

A324325 Number of non-crossing multiset partitions of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 5, 9, 7, 7, 11, 11, 12, 16, 14, 15, 26, 22, 21, 29, 19, 30, 33, 31, 30, 66, 38, 42, 52, 56, 42, 47, 45, 57, 82, 77, 67, 77, 67, 101, 98, 135, 64, 137, 97, 176, 104, 109, 109, 118, 105, 231, 213, 97, 127, 181, 139, 297, 173, 385, 195, 269

Offset: 1

Gus Wiseman, Feb 22 2019

nonn

This multiset (row n of A305936) is generally not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

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

			The a(16) = 14 non-crossing multiset partitions of the multiset {1,2,3,4}:
  {{1,2,3,4}}
  {{1},{2,3,4}}
  {{2},{1,3,4}}
  {{3},{1,2,4}}
  {{4},{1,2,3}}
  {{1,2},{3,4}}
  {{1,4},{2,3}}
  {{1},{2},{3,4}}
  {{1},{3},{2,4}}
  {{1},{4},{2,3}}
  {{2},{3},{1,4}}
  {{2},{4},{1,3}}
  {{3},{1,2},{4}}
  {{1},{2},{3},{4}}
Missing from this list is {{1,3},{2,4}}.

Cf. A000108, A001055, A001970, A016098, A054726, A099947, A181821, A305936, A306438, A318284, A318285.

Cf. A324167, A324168, A324169, A324170, A324171, A324324, A324326.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
nonXQ[stn_]:=!MatchQ[stn,{_,{_,x_,_,y_,_},_,{_,z_,_,t_,_},_}/;x

a(n) + A324326(n) = A318284(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

A324326 Number of crossing multiset partitions of a multiset whose multiplicities are the prime indices of n.

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A326291 Number of unsortable factorizations of n.

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A324325 Number of non-crossing multiset partitions of a multiset whose multiplicities are the prime indices of n.

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A326277 Number of crossing normal multiset partitions of weight n.

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