A322441 -id:A322441 - cp's OEIS frontend

A320632 Numbers k such that there exists a pair of factorizations of k into factors > 1 where no factor of one divides any factor of the other.

36, 60, 72, 84, 90, 100, 108, 120, 126, 132, 140, 144, 150, 156, 168, 180, 196, 198, 200, 204, 210, 216, 220, 225, 228, 234, 240, 252, 260, 264, 270, 276, 280, 288, 294, 300, 306, 308, 312, 315, 324, 330, 336, 340, 342, 348, 350, 360, 364, 372, 378, 380, 390

Offset: 1

Gus Wiseman, Dec 09 2018

nonn

Positions of nonzero terms in A322437 or A322438.
Mats Granvik has conjectured that these are all the positive integers k such that sigma_0(k) - 2 > (bigomega(k) - 1) * omega(k), where sigma_0 = A000005, omega = A001221, and bigomega = A001222. - Gus Wiseman, Nov 12 2019
Numbers with more semiprime divisors than prime divisors. - Wesley Ivan Hurt, Jun 10 2021

			An example of such a pair for 36 is (4*9)|(6*6).

Antti Karttunen, Table of n, a(n) for n = 1..23437
Christophe Cordero, Factorizations of Cyclic Groups and Bayonet Codes, arXiv:2301.13566 [math.CO], 2023, p. 20.

Cf. A001055, A050336, A285572, A303362, A305149, A305193, A317144, A322435, A322437, A322439, A322440, A322441, A322442.
The following are additional cross-references relating to Granvik's conjecture.
bigomega(n) * omega(n) is A113901(n).
(bigomega(n) - 1) * omega(n) is A307409(n).
sigma_0(n) - bigomega(n) * omega(n) is A328958(n).
sigma_0(n) - 2 - (omega(n) - 1) * nu(n) is A328959(n).
Cf. A060687, A070175, A124010, A323023, A328956, A328957, A328960, A328961, A328963.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[100],Select[Subsets[facs[#],{2}],And[!Or@@Divisible@@@Tuples[#],!Or@@Divisible@@@Reverse/@Tuples[#]]&]!={}&]

factorizations(n, m=n, f=List([]), z=List([])) = if(1==n, listput(z,Vec(f)); z, my(newf); fordiv(n, d, if((d>1)&&(d<=m), newf = List(f); listput(newf,d); z = factorizations(n/d, d, newf, z))); (z));
is_ndf_pair(fac1,fac2) = { for(i=1,#fac1,for(j=1,#fac2,if(!(fac1[i]%fac2[j])||!(fac2[j]%fac1[i]),return(0)))); (1); };
has_at_least_one_ndfpair(z) = { for(i=1,#z,for(j=i+1,#z,if(is_ndf_pair(z[i],z[j]),return(1)))); (0); };
isA320632(n) = has_at_least_one_ndfpair(Vec(factorizations(n))); \\ Antti Karttunen, Dec 10 2020

A322439 Number of ordered pairs of integer partitions of n where no part of the first is greater than any part of the second.

Original entry on oeis.org

1, 1, 3, 5, 11, 15, 33, 42, 82, 114, 195, 258, 466, 587, 954, 1317, 2021, 2637, 4124, 5298, 7995, 10565, 15075, 19665, 28798, 36773, 51509, 67501, 93060, 119299, 165589, 209967, 285535, 366488, 487536, 622509, 833998, 1048119, 1380410, 1754520, 2291406, 2876454

Offset: 0

Gus Wiseman, Dec 08 2018

nonn

			The a(5) = 15 pairs of integer partitions:
      (5)|(5)
     (41)|(5)
     (32)|(5)
    (311)|(5)
    (221)|(5)
    (221)|(32)
   (2111)|(5)
   (2111)|(32)
  (11111)|(5)
  (11111)|(41)
  (11111)|(32)
  (11111)|(311)
  (11111)|(221)
  (11111)|(2111)
  (11111)|(11111)

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A026794, A026820, A265947, A285573, A317144, A318915, A322435, A322436, A322440, A322441, A322442, A362051.

g:= proc(n, i) option remember; `if`(n=0 or i=1, 1,
      g(n, i-1) +g(n-i, min(i, n-i)))
    end:
b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i>n, 0, b(n, i+1)+b(n-i, i)))
    end:
a:= proc(n) option remember; `if`(n=0, 1,
      add(g(n, i)*b(n-i, i), i=1..n))
    end:
seq(a(n), n=0..50);  # Alois P. Heinz, Dec 09 2018

Table[Length[Select[Tuples[IntegerPartitions[n],2],Max@@First[#]<=Min@@Last[#]&]],{n,20}]
(* Second program: *)
g[n_, i_] := g[n, i] = If[n == 0 || i == 1, 1, g[n, i - 1] + g[n - i, Min[i, n - i]]];
b[n_, i_] := b[n, i] = If[n == 0, 1, If[i>n, 0, b[n, i+1] + b[n-i, i]]];
a[n_] := a[n] = If[n == 0, 1, Sum[g[n, i]*b[n - i, i], {i, 1, n}]];
a /@ Range[0, 50] (* Jean-François Alcover, May 17 2021, after Alois P. Heinz *)

a(n) = Sum_{k = 1..n} A026820(n,k) * A026794(n,k).

a(n) = A000041(2n) - A362051(n) for n>=1. - Alois P. Heinz, Apr 27 2023

A322442 Number of pairs of set partitions of {1,...,n} where every block of one is a subset or superset of some block of the other.

Original entry on oeis.org

1, 1, 4, 25, 195, 1894, 22159, 303769, 4790858, 85715595, 1720097275, 38355019080, 942872934661, 25383601383937, 744118939661444, 23635548141900445, 809893084668253151, 29822472337116844174, 1175990509568611058299, 49504723853840395163221, 2218388253903492656783562

Offset: 0

Gus Wiseman, Dec 08 2018

nonn

			The a(3) = 25 pairs of set partitions (these are actually all pairs of set partitions of {1,2,3}):
  (1)(2)(3)|(1)(2)(3)
  (1)(2)(3)|(1)(23)
  (1)(2)(3)|(12)(3)
  (1)(2)(3)|(13)(2)
  (1)(2)(3)|(123)
    (1)(23)|(1)(2)(3)
    (1)(23)|(1)(23)
    (1)(23)|(12)(3)
    (1)(23)|(13)(2)
    (1)(23)|(123)
    (12)(3)|(1)(2)(3)
    (12)(3)|(1)(23)
    (12)(3)|(12)(3)
    (12)(3)|(13)(2)
    (12)(3)|(123)
    (13)(2)|(1)(2)(3)
    (13)(2)|(1)(23)
    (13)(2)|(12)(3)
    (13)(2)|(13)(2)
    (13)(2)|(123)
      (123)|(1)(2)(3)
      (123)|(1)(23)
      (123)|(12)(3)
      (123)|(13)(2)
      (123)|(123)
Non-isomorphic representatives of the pairs of set partitions of {1,2,3,4} for which the condition fails:
    (12)(34)|(13)(24)
    (12)(34)|(1)(3)(24)
  (1)(2)(34)|(13)(24)

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

Cf. A000110, A000258, A001247, A008277, A059849, A060639, A181939, A318393, A322435, A322437, A322439, A322440, A322441.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
costabQ[s_,t_]:=And@@Cases[s,x_:>Select[t,SubsetQ[x,#]||SubsetQ[#,x]&]!={}];
Table[Length[Select[Tuples[sps[Range[n]],2],And[costabQ@@#,costabQ@@Reverse[#]]&]],{n,5}]

F(x)={my(bell=(exp(y*(exp(x) - 1))  )); subst(serlaplace( serconvol(bell, bell)), y, exp(exp(x) - 1)-1)}
seq(n) = {my(x=x + O(x*x^n)); Vec(serlaplace( exp( 2*exp(exp(x) - 1) - exp(x) - 1) * F(x) ))} \\ Andrew Howroyd, Jan 19 2024

E.g.f.: exp(exp(x)-1) * (2*B(x) - 1) where B(x) is the e.g.f. of A319884. - Andrew Howroyd, Jan 19 2024

a(8) onwards from Andrew Howroyd, Jan 19 2024

A322435 Number of pairs of factorizations of n into factors > 1 where no factor of the second divides any factor of the first.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 3, 0, 1, 1, 5, 0, 3, 0, 3, 1, 1, 0, 7, 1, 1, 2, 3, 0, 4, 0, 7, 1, 1, 1, 15, 0, 1, 1, 7, 0, 4, 0, 3, 3, 1, 0, 16, 1, 3, 1, 3, 0, 7, 1, 7, 1, 1, 0, 18, 0, 1, 3, 16, 1, 4, 0, 3, 1, 4, 0, 32, 0, 1, 3, 3, 1, 4, 0, 16, 5, 1, 0, 18, 1

Offset: 1

Gus Wiseman, Dec 08 2018

nonn

			The a(36) = 15 pairs of factorizations:
  (2*2*3*3)|(4*9)
  (2*2*3*3)|(6*6)
  (2*2*3*3)|(36)
    (2*2*9)|(6*6)
    (2*2*9)|(36)
    (2*3*6)|(4*9)
    (2*3*6)|(36)
     (2*18)|(36)
    (3*3*4)|(6*6)
    (3*3*4)|(36)
     (3*12)|(36)
      (4*9)|(6*6)
      (4*9)|(36)
      (6*6)|(4*9)
      (6*6)|(36)

Cf. A000688, A001055, A050336, A265947, A294068, A305149, A305150, A305193, A317144, A322436, A322437, A322438, A322439, A322440, A322441, A322442.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[Tuples[facs[n],2],!Or@@Divisible@@@Tuples[#]&]],{n,100}]

A322437 Number of unordered pairs of factorizations of n into factors > 1 where no factor of one divides any factor of the other.

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, 0, 0, 0, 0, 1, 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, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2

Offset: 1

Gus Wiseman, Dec 08 2018

nonn

First differs from A322438 at a(144) = 3, A322438(144) = 4.
From Antti Karttunen, Dec 11 2020: (Start)
Zeros occur on numbers that are either of the form p^k, or q * p^k, or p*q*r, for some primes p, q, r, and exponent k >= 0. [Note also that in all these cases, when x > 1, A307408(x) = 2+A307409(x) = 2 + (A001222(x) - 1)*A001221(x) = A000005(x)].
Proof:
  It is easy to see that for such numbers it is not possible to obtain two such distinct factorizations, that no factor of the other would not divide some factor of the other.
  Conversely, the complement set of above is formed of such composites n that have at least one unitary divisor that is either of the form
  (1) p^x * q^y, with x, y >= 2,
  or
  (2) p^x * q^y * r^z, with x >= 2, and y, z >= 1,
  or
  (3) p^x * q^y * r^z * s^w, with x, y, z, w >= 1,
  where p, q, r, s are distinct primes. Let's indicate with C the remaining portion of k coprime to p, q, r and s (which could be also 1). Then in case (1) we can construct two factorizations, the first having factors (p*q*C) and (p^(x-1) * q^(y-1)), and the second having factors (p^x * C) and (q^y) that are guaranteed to satisfy the condition that no factor in the other factorization divides any of the factors of the other factorization. For case (2) pairs like {(p * q^y * C), (p^(x-1) * r^z)} and {(p^x * C), (q^y * r^z)}, and for case (3) pairs like {(p^x * q^y * C), (r^z * s^w)} and {(p^x * r^z * C), (q^y * s^w)} offer similar examples, therefore a(n) > 0 for all such cases.
(End)

			The a(120) = 2 pairs of such factorizations:
   (6*20)|(8*15)
   (8*15)|(10*12)
The a(144) = 3 pairs of factorizations:
   (6*24)|(9,16)
   (8*18)|(12*12)
   (9*16)|(12*12)
The a(210) = 3 pairs of factorizations:
   (6*35)|(10*21)
   (6*35)|(14*15)
  (10*21)|(14*15)
[Note that 210 is the first squarefree number obtaining nonzero value]
The a(240) = 4 pairs of factorizations:
   (6*40)|(15*16)
   (8*30)|(12*20)
  (10*24)|(15*16)
  (12*20)|(15*16)
The a(1728) = 14 pairs of factorizations:
    (6*6*48)|(27*64)
   (6*12*24)|(27*64)
     (6*288)|(27*64)
    (8*8*27)|(12*12*12)
  (12*12*12)|(27*64)
  (12*12*12)|(32*54)
    (12*144)|(27*64)
    (12*144)|(32*54)
    (16*108)|(24*72)
     (18*96)|(27*64)
     (24*72)|(27*64)
     (24*72)|(32*54)
     (27*64)|(36*48)
     (32*54)|(36*48)

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A000688, A001055, A285572, A285573, A294068, A303362, A304713, A305149, A305150, A305193, A316476, A317144, A322435, A322436, A322441, A322442, A339626.

Cf. also comments A307408 and A307409.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[Subsets[facs[n],{2}],And[!Or@@Divisible@@@Tuples[#],!Or@@Divisible@@@Reverse/@Tuples[#]]&]],{n,100}]

factorizations(n, m=n, f=List([]), z=List([])) = if(1==n, listput(z,Vec(f)); z, my(newf); fordiv(n, d, if((d>1)&&(d<=m), newf = List(f); listput(newf,d); z = factorizations(n/d, d, newf, z))); (z));
is_ndf_pair(fac1,fac2) = { for(i=1,#fac1,for(j=1,#fac2,if(!(fac1[i]%fac2[j])||!(fac2[j]%fac1[i]),return(0)))); (1); };
number_of_ndfpairs(z) = sum(i=1,#z,sum(j=i+1,#z,is_ndf_pair(z[i],z[j])));
A322437(n) = number_of_ndfpairs(Vec(factorizations(n))); \\ Antti Karttunen, Dec 10 2020

For n > 0, a(A002110(n)) = A322441(n)/2 = A339626(n). - Antti Karttunen, Dec 10 2020

Data section extended up to a(120) and more examples added by Antti Karttunen, Dec 10 2020

A322436 Number of pairs of factorizations of n into factors > 1 where no factor of the second properly divides any factor of the first.

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 5, 3, 3, 1, 8, 1, 3, 3, 11, 1, 8, 1, 8, 3, 3, 1, 18, 3, 3, 5, 8, 1, 12, 1, 15, 3, 3, 3, 31, 1, 3, 3, 18, 1, 12, 1, 8, 8, 3, 1, 39, 3, 8, 3, 8, 1, 18, 3, 18, 3, 3, 1, 42, 1, 3, 8, 33, 3, 12, 1, 8, 3, 12, 1, 67, 1, 3, 8, 8, 3, 12, 1, 39, 11

Offset: 1

Gus Wiseman, Dec 08 2018

nonn

			The a(12) = 8 pairs of factorizations:
  (2*2*3)|(2*2*3)
  (2*2*3)|(2*6)
  (2*2*3)|(3*4)
  (2*2*3)|(12)
    (2*6)|(12)
    (3*4)|(3*4)
    (3*4)|(12)
     (12)|(12)

Cf. A000688, A001055, A050336, A265947, A281113, A294068, A303386, A305149, A305150, A305193, A317144, A322435, A322439, A322440, A322441, A322442.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
divpropQ[x_,y_]:=And[x!=y,Divisible[x,y]];
Table[Length[Select[Tuples[facs[n],2],!Or@@divpropQ@@@Tuples[#]&]],{n,100}]

A319884 Number of unordered pairs of set partitions of {1,...,n} where every block of one is a proper subset or proper superset of some block of the other.

Original entry on oeis.org

1, 0, 1, 7, 50, 481, 5667, 78058, 1238295, 22314627, 451354476, 10148011215, 251584513215, 6831141750512, 201976943666357, 6470392653260939, 223595676728884394, 8302299221314559877, 330075531021130110015, 14006780163088113914026, 632606447496264724088803

Offset: 0

Gus Wiseman, Dec 09 2018

nonn

			The a(3) = 7 pairs of set partitions:
  (1)(2)(3)|(123)
    (1)(23)|(12)(3)
    (1)(23)|(13)(2)
    (1)(23)|(123)
    (12)(3)|(13)(2)
    (12)(3)|(123)
    (13)(2)|(123)

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

Cf. A000110, A000258, A001247, A008277, A059849, A060639, A181939, A322435, A322436, A322437, A322438, A322439, A322440, A322441, A322442.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
costabstrQ[s_,t_]:=And@@Cases[s,x_:>Select[t,x!=#&&(SubsetQ[x,#]||SubsetQ[#,x])&]!={}];
Table[Length[Select[Subsets[sps[Range[n]],{2}],And[costabstrQ@@#,costabstrQ@@Reverse[#]]&]],{n,5}]

F(x)={my(bell=(exp(y*(exp(x) - 1))  )); subst(serlaplace( serconvol(bell, bell)), y, exp(exp(x) - 1)-1)}
seq(n) = {my(x=x + O(x*x^n)); Vec(serlaplace( 1 + exp( 2*(exp(exp(x) - 1) - exp(x)) ) * F(x) )/2)} \\ Andrew Howroyd, Jan 19 2024

\\ 2nd prog, following formula - slightly slower
D(n,y) = (exp(2*y)/(1 + y)^2) * sum(k=0,n, x^k*sum(j=0, k, stirling(k,j,2) * y^j)^2/k!, O(x*x^n))
seq(n) = Vec(serlaplace((1/2)*(1 + D(n, exp(exp(x + O(x*x^n)) - 1) - 1)))) \\ Andrew Howroyd, Jan 20 2024

E.g.f.: (1/2)*(1 + D(x, exp(exp(x) - 1) - 1) ) where D(x,y) = (exp(2*y)/(1 + y)^2) * Sum_{k>=0} x^k*(Sum_{j=0..k} Stirling2(k,j)*y^j)^2/k!. - Andrew Howroyd, Jan 20 2024

a(8) onwards from Andrew Howroyd, Jan 19 2024

A152525 a(n) is the number of unordered pairs of disjoint set partitions of an n-element set.

Original entry on oeis.org

0, 0, 1, 7, 65, 811, 12762, 244588, 5574956, 148332645, 4538695461, 157768581675, 6167103354744, 268758895112072, 12961171404183498, 687270616305277589, 39843719438374998543, 2512873126513271758171, 171643113190082528007702, 12647168303374365311984284

Offset: 0

David Pasino, Dec 06 2008, Dec 08 2008

nonn

			From _Gus Wiseman_, Dec 09 2018: (Start)
The a(3) = 7 unordered pairs:
  {{1},{2},{3}}| {{1,2,3}}
   {{1},{2,3}} |{{1,2},{3}}
   {{1},{2,3}} |{{1,3},{2}}
   {{1,2},{3}} |{{1,3},{2}}
   {{1},{2,3}} | {{1,2,3}}
   {{1,2},{3}} | {{1,2,3}}
   {{1,3},{2}} | {{1,2,3}}
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..200
Frank Ruskey and Jennifer Woodcock, The Rand and block distances of pairs of set partitions, Combinatorial algorithms, 287-299, Lecture Notes in Comput. Sci., 7056, Springer, Heidelberg, 2011.
Frank Ruskey, Jennifer Woodcock and Yuji Yamauchi, Counting and computing the Rand and block distances of pairs of set partitions, Journal of Discrete Algorithms, Volume 16, October 2012, Pages 236-248. - From _N. J. A. Sloane_, Oct 03 2012

Cf. A000110, A000258, A001247, A008277, A048993, A059849, A060639, A181939, A193297, A318393, A322441, A322442, A320768.

a:= n-> add(binomial(n,k)*binomial(combinat[bell](k),2)*
        add(Stirling2(n-k,j)*(-1)^j, j=0..n-k), k=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, May 27 2018

Array[Sum[Binomial[#, k] Sum[(-1)^j*StirlingS2[# - k, j], {j, 0, # - k}] Binomial[BellB@ k, 2], {k, 0, #}] &, 20, 0] (* Michael De Vlieger, May 27 2018 *)

a000110(n) = polcoeff( sum( k=0, n, prod( i=1, k, x / (1 - i*x)), x^n * O(x)), n);
a(n) = sum(k=0, n, binomial(n,k) * sum(j=0, n-k, (-1)^j*stirling(n-k,j, 2) * binomial(a000110(k),2))); \\ Michel Marcus, May 27 2018

a(n) = Sum_{k=0..n} binomial(n,k) * (Sum_{j=0..n-k} (-1)^j*A048993(n-k,j)) * binomial(A000110(k),2).
That is, summed on k from 0 to n, the number of k-element subsets of an n-element set, times the alternating sum of row n-k of Stirling2 numbers starting with +S(n-k, 0), times the number of pairs of partitions of k elements.
Obtained by inverting (binomial(A000110(n), 2)) = (Sum_{k=0..n} binomial(n,k)*A000110(n-k)*a(k)), which in turn is gotten by considering that a pair of conjoint partitions is gotten by choosing a partition of a subset and then choosing a pair of disjoint partitions of the complement.

A339626 a(n) = A322437(A002110(n)).

Original entry on oeis.org

0, 0, 0, 0, 3, 30, 315, 4830, 96453, 2296350

Offset: 0

Antti Karttunen, Dec 10 2020

For n > 0, a(n) gives the number of unordered pairs of set partitions of {1,...,n} where no block of the other is a subset (or equal) to any block of the other. See A322441.

			The a(4) = 3 such (unordered) pairs of set partitions of {1,2,3,4} are:
  {{1,2},{3,4}}|{{1,3},{2,4}}
  {{1,2},{3,4}}|{{1,4},{2,3}}
  {{1,3},{2,4}}|{{1,4},{2,3}}.

Cf. A002110, A322437, A322441.

Block[{f}, f[n_] := If[n <= 1, {{}}, Join @@ Table[Map[Prepend[#, d] &, Select[f[n/d], Min @@ # >= d &]], {d, Rest[Divisors[n]]}]]; Map[Length[Select[Subsets[f[#], {2}], And[! Or @@ Divisible @@@ #, ! Or @@ Divisible @@@ Reverse /@ #] &@ Tuples[#] &]] &, FoldList[Times, 1, Prime@ Range@ 7]] ] (* Michael De Vlieger, Dec 10 2020, after Gus Wiseman at A322437 *)

For n > 0, a(n) = A322441(n)/2.

cp's OEIS Frontend

A320632 Numbers k such that there exists a pair of factorizations of k into factors > 1 where no factor of one divides any factor of the other.

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A322439 Number of ordered pairs of integer partitions of n where no part of the first is greater than any part of the second.

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A322442 Number of pairs of set partitions of {1,...,n} where every block of one is a subset or superset of some block of the other.

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A322435 Number of pairs of factorizations of n into factors > 1 where no factor of the second divides any factor of the first.

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A322437 Number of unordered pairs of factorizations of n into factors > 1 where no factor of one divides any factor of the other.

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A322436 Number of pairs of factorizations of n into factors > 1 where no factor of the second properly divides any factor of the first.

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A319884 Number of unordered pairs of set partitions of {1,...,n} where every block of one is a proper subset or proper superset of some block of the other.

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A152525 a(n) is the number of unordered pairs of disjoint set partitions of an n-element set.

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