A322436 -id:A322436 - cp's OEIS frontend

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

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

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

A322438 Number of unordered pairs of factorizations of n into factors > 1 where no factor of one properly divides any factor of the other.

Original entry on oeis.org

Offset: 1

Gus Wiseman, Dec 08 2018

nonn

First differs from A322437 at a(144) = 4, A322437(144) = 3.

First differs from A379958 at a(120) = 2, A379958(120) = 1.

			The a(240) = 5 pairs of factorizations::
  (4*4*15)|(4*6*10)
    (6*40)|(15*16)
    (8*30)|(12*20)
   (10*24)|(15*16)
   (12*20)|(15*16)

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

Cf. A001055, A285572, A285573, A303362, A303386, A304713, A305149, A305150, A305193, A316476, A317144, A322435, A322436, A322437, A322442, A379958.

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[Subsets[facs[n],{2}],And[!Or@@divpropQ@@@Tuples[#],!Or@@divpropQ@@@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_proper_ndf_pair(fac1,fac2) = { for(i=1,#fac1,for(j=1,#fac2,if((fac1[i]!=fac2[j]) && (!(fac1[i]%fac2[j]) || !(fac2[j]%fac1[i])),return(0)))); (1); };
number_of_proper_ndfpairs(z) = sum(i=1,#z,sum(j=i+1,#z,is_proper_ndf_pair(z[i],z[j])));
A322438(n) = number_of_proper_ndfpairs(Vec(factorizations(n))); \\ Antti Karttunen, Jan 24 2025

Data section extended up to a(144) by Antti Karttunen, Jan 24 2025

A322440 Number of pairs of integer partitions of n where every part of the first is less than every part of the second.

Original entry on oeis.org

1, 0, 1, 2, 5, 7, 16, 20, 40, 55, 97, 124, 235, 287, 482, 654, 1033, 1318, 2137, 2676, 4157, 5439, 7891, 10144, 15280, 19171, 27336, 35652, 49756, 63150, 89342, 111956, 154400, 197413, 264572, 336082, 456724, 568932, 756065, 959566, 1261803, 1576355, 2078267

Offset: 0

Gus Wiseman, Dec 08 2018

nonn

			The a(5) = 16 pairs of integer partitions:
      (51)|(6)
      (42)|(6)
     (411)|(6)
      (33)|(6)
     (321)|(6)
    (3111)|(6)
     (222)|(6)
     (222)|(33)
    (2211)|(6)
    (2211)|(33)
   (21111)|(6)
   (21111)|(33)
  (111111)|(6)
  (111111)|(42)
  (111111)|(33)
  (111111)|(222)

Cf. A265947, A317144, A318915, A322435, A322436, A322439.

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, min(n-i, i))*b(n, i+1), 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[#]n, 0, b[n, i+1] + b[n-i, i]]];
a[n_] := a[n] = If[n==0, 1, Sum[g[n-i, Min[n-i, i]]*b[n, i+1], {i, 1, n}]];
a /@ Range[0, 50] (* Jean-François Alcover, May 10 2021, after Alois P. Heinz *)

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

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

cp's OEIS Frontend

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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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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A322438 Number of unordered pairs of factorizations of n into factors > 1 where no factor of one properly divides any factor of the other.

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Mathematica

PARI

Extensions

A322440 Number of pairs of integer partitions of n where every part of the first is less than every part of the second.

Views

Author

Keywords

Examples

Crossrefs

Programs

Maple

Mathematica

Formula

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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Examples

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Mathematica

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PARI

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Extensions