A116539 -id:A116539 - cp's OEIS frontend

A382077 Number of integer partitions of n that can be partitioned into a set of sets.

1, 1, 1, 2, 3, 5, 6, 9, 13, 17, 25, 33, 44, 59, 77, 100, 134, 171, 217, 283, 361, 449, 574, 721, 900, 1126, 1397, 1731, 2143, 2632, 3223, 3961, 4825, 5874, 7131, 8646, 10452, 12604, 15155, 18216, 21826, 26108, 31169, 37156, 44202, 52492, 62233, 73676, 87089, 102756, 121074

Offset: 0

Gus Wiseman, Mar 18 2025

nonn

First differs from A240306 at a(14) = 76, A240306(14) = 77.

First differs from A381992 at a(17) = 171, A381992(17) = 170.

			For y = (3,2,2,2,1,1,1), we have the multiset partition {{1},{2},{1,2},{1,2,3}}, so y is counted under a(12).
The a(1) = 1 through a(8) = 13 partitions:
  (1)  (2)  (3)    (4)      (5)      (6)        (7)        (8)
            (2,1)  (3,1)    (3,2)    (4,2)      (4,3)      (5,3)
                   (2,1,1)  (4,1)    (5,1)      (5,2)      (6,2)
                            (2,2,1)  (3,2,1)    (6,1)      (7,1)
                            (3,1,1)  (4,1,1)    (3,2,2)    (3,3,2)
                                     (2,2,1,1)  (3,3,1)    (4,2,2)
                                                (4,2,1)    (4,3,1)
                                                (5,1,1)    (5,2,1)
                                                (3,2,1,1)  (6,1,1)
                                                           (3,2,2,1)
                                                           (3,3,1,1)
                                                           (4,2,1,1)
                                                           (3,2,1,1,1)

Factorizations of this type are counted by A050345.
More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
Normal multiset partitions of this type are counted by A116539.
The MM-numbers of these multiset partitions are A302494.
Twice-partitions of this type are counted by A358914.
For distinct block-sums instead of blocks we have A381992, ranked by A382075.
The complement is counted by A382078, unique A382079.
These partitions are ranked by A382200, complement A293243.
For normal multisets instead of integer partitions we have A382214, complement A292432.
A000041 counts integer partitions, strict A000009.
A050320 counts multiset partitions of prime indices into sets.
A050326 counts multiset partitions of prime indices into distinct sets.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A292444, A293511, A299202, A317142, A381441, A381454, A381717, A381718, A381870, A381990.

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]]]];
Table[Length[Select[IntegerPartitions[n], Length[Select[mps[#],UnsameQ@@#&&And@@UnsameQ@@@#&]]>0&]],{n,0,9}]

a(21)-a(50) from Bert Dobbelaere, Mar 29 2025

A181230 Square array T(m,n) giving the number of m X n (0,1)-matrices with pairwise distinct rows and pairwise distinct columns.

Original entry on oeis.org

2, 2, 2, 0, 10, 0, 0, 24, 24, 0, 0, 24, 264, 24, 0, 0, 0, 1608, 1608, 0, 0, 0, 0, 6720, 33864, 6720, 0, 0, 0, 0, 20160, 483840, 483840, 20160, 0, 0, 0, 0, 40320, 5644800, 19158720, 5644800, 40320, 0, 0, 0, 0, 40320, 57415680, 595506240, 595506240, 57415680, 40320

Offset: 1

R. H. Hardin, Oct 10 2010

			Table starts
.2..2.....0...........0...............0..................0
.2.10....24..........24...............0..................0
.0.24...264........1608............6720..............20160
.0.24..1608.......33864..........483840............5644800
.0..0..6720......483840........19158720..........595506240
.0..0.20160.....5644800.......595506240........44680224960
.0..0.40320....57415680.....16388749440......2881362718080
.0..0.40320...518676480....418910083200....172145618789760
.0..0.....0..4151347200..10136835072000...9841604944066560
.0..0.....0.29059430400.233811422208000.546156941728204800

R. H. Hardin, Table of n, a(n) for n=1..180
MathOverflow, Number of matrices with no repeated columns or rows

Cf. A088310 (diagonal), A181231, A181232, A181233 (subdiagonals).

Binary matrices with distinct rows and columns, various versions: A059202, A088309, A088310, A088616, A089673, A089674, A093466, A094000, A094223, A116532, A116539, A181230, A259763

T(m,n) = Sum_{i=0..n} Sum_{j=0..m} stirling1(n,i) * stirling1(m,j) * 2^(i*j) = n! * Sum_{j=0..m} stirling1(m,j) * binomial(2^j,n) = m! * Sum_{i=0..n} stirling1(n,i) * binomial(2^i,m). - Max Alekseyev, Jun 18 2016

T(m,n) = A059084(m,n) * n!.

A059202 Triangle T(n,m) of numbers of m-block T_0-covers of a labeled n-set, m = 0..2^n - 1.

Original entry on oeis.org

1, 0, 1, 0, 0, 3, 1, 0, 0, 3, 29, 35, 21, 7, 1, 0, 0, 0, 140, 1015, 2793, 4935, 6425, 6435, 5005, 3003, 1365, 455, 105, 15, 1, 0, 0, 0, 420, 13965, 126651, 661801, 2533135, 7792200, 20085000, 44307120, 84651840, 141113700, 206251500, 265182300

Offset: 0

Vladeta Jovovic, Goran Kilibarda, Jan 18 2001

A cover of a set is a T_0-cover if for every two distinct points of the set there exists a member (block) of the cover containing one but not the other point.

Also, T(n,m) is the number of n X m (0,1)-matrices with pairwise distinct nonzero columns and pairwise distinct nonzero rows, up to permutation of columns.

			[1],
[0,1],
[0,0,3,1],
[0,0,3,29,35,21,7,1],
...
There are 35 4-block T_0-covers of a labeled 3-set.

Robert Israel, Table of n, a(n) for n = 0..4094 (rows 0 to 11, flattened).
T_0-covers of a labeled 3-set

Cf. A059201 (row sums), A059203 (column sums), A094000 (main diagonal).
Cf. A059084, A059085, A059086, A059087, A059088, A059089, A181230.
Binary matrices with distinct rows and columns, various versions: A059202, A088309, A088310, A088616, A089673, A089674, A093466, A094000, A094223, A116532, A116539, A181230, A259763

with(combinat): for n from 0 to 10 do for m from 0 to 2^n-1 do printf(`%d,`,(1/m!)*sum(stirling1(m+1,i)*product(2^(i-1)-1-j, j=0..n-1), i=1..m+1)) od: od:

T[n_, m_] = Sum[ StirlingS1[n + 1, i + 1]*Binomial[2^i - 1, m], {i, 0, n}]; Table[T[n, m], {n, 0, 5}, {m, 0, 2^n - 1}] (* G. C. Greubel, Dec 28 2016 *)

T(n, m) = (1/m!)*Sum_{1..m + 1} stirling1(m + 1, i)*[2^(i - 1) - 1]_n, where [k]_n := k*(k - 1)*...*(k - n + 1), [k]_0 = 1.
E.g.f: Sum((1+x)^(2^n-1)*log(1+y)^n/n!, n=0..infinity)/(1+y). - Vladeta Jovovic, May 19 2004
Also T(n, m) = Sum_{i=0..n} Stirling1(n+1, i+1)*binomial(2^i-1, m). - Vladeta Jovovic, Jun 04 2004
T(n,m) = A181230(n,m)/m! - n*T(n-1,m) - T(n,m-1) - n*T(n-1,m-1). - Max Alekseyev, Dec 11 2017

More terms from James Sellers, Jan 24 2001

A088310 Number of n X n (0,1)-matrices with all rows distinct and all columns distinct.

Original entry on oeis.org

1, 2, 10, 264, 33864, 19158720, 44680224960, 413586858182400, 14960200449325582080, 2109063823453947981680640, 1162864344149083760773678387200, 2520991223487759548686737154649702400, 21598422878151131130336454273775859841843200, 734233037731110118818452425552296701963294284185600

Offset: 0

N. J. A. Sloane, Nov 07 2003

nonn

			a(2) = 10: 00/01, 00/10, 01/00, 01/10, 01/11, 10/00, 10/01, 10/11, 11/01, 11/10.

G. C. Greubel, Table of n, a(n) for n = 0..57

Cf. A088229, A088309.

Binary matrices with distinct rows and columns, various versions: A059202, A088309, A088310, A088616, A089673, A089674, A093466, A094000, A094223, A116532, A116539, A181230, A259763

A088310:= func< n | Factorial(n)*(&+[Binomial(2^k,n)*StirlingFirst(n,k): k in [0..n]]) >;
[A088310(n): n in [0..30]]; // G. C. Greubel, Dec 14 2022

Table[n!*Sum[StirlingS1[n, k]*Binomial[2^k,n], {k, 0, n}], {n, 0, 15}] (* Vaclav Kotesovec, Jul 02 2016 *)

@CachedFunction
def A088310(n): return (-1)^n*factorial(n)*sum((-1)^k*binomial(2^k,n)*stirling_number1(n,k) for k in (0..n))
[A088310(n) for n in range(31)] # G. C. Greubel, Dec 14 2022

a(n) = n! * Sum_{k=0..n} Stirling1(n, k)*binomial(2^k, n). - Vladeta Jovovic, Nov 07 2003
a(n) = Sum_{i=0..n} Sum_{j=0..n} stirling1(n, i) * stirling1(n, j) * 2^(i*j). - Max Alekseyev, Nov 07 2003
a(n) ~ 2^(n^2). - Vaclav Kotesovec, Jul 02 2016
a(n) = A181230(n,n).

Suggested by Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 06 2003

a(0)-a(5) from W. Edwin Clark, Nov 07 2003

A116532 Number of singular n X n rational {0,1}-matrices with no zero rows or columns and with all rows distinct, up to permutation of rows.

Original entry on oeis.org

0, 0, 3, 285, 50820, 23551920, 31898503077, 134251404794199

Offset: 1

Vladeta Jovovic, Apr 03 2006

Cf. A000410, A116506, A116527.

Binary matrices with distinct rows and columns, various versions: A059202, A088309, A088310, A088616, A089673, A089674, A093466, A094000, A094223, A116532, A116539, A181230, A259763

a(n) = A054780(n) - A088389(n).

A382078 Number of integer partitions of n that cannot be partitioned into a set of sets.

Original entry on oeis.org

0, 0, 1, 1, 2, 2, 5, 6, 9, 13, 17, 23, 33, 42, 58, 76, 97, 126, 168, 207, 266, 343, 428, 534, 675, 832, 1039, 1279, 1575, 1933, 2381, 2881, 3524, 4269, 5179, 6237, 7525, 9033, 10860, 12969, 15512, 18475, 22005, 26105, 30973, 36642, 43325, 51078, 60184, 70769, 83152

Offset: 0

Gus Wiseman, Mar 18 2025

nonn

First differs from A240309 at a(11) = 23, A240309(11) = 25.

First differs from A381990 at a(17) = 126, A381990(17) = 127.

			The partition y = (2,2,1,1,1) can be partitioned into sets in the following ways:
  {{1},{1,2},{1,2}}
  {{1},{1},{2},{1,2}}
  {{1},{1},{1},{2},{2}}
But none of these is itself a set, so y is counted under a(7).
The a(2) = 1 through a(8) = 9 partitions:
  (11)  (111)  (22)    (2111)   (33)      (2221)     (44)
               (1111)  (11111)  (222)     (4111)     (2222)
                                (3111)    (22111)    (5111)
                                (21111)   (31111)    (22211)
                                (111111)  (211111)   (41111)
                                          (1111111)  (221111)
                                                     (311111)
                                                     (2111111)
                                                     (11111111)

More on set multipartitions: A089259, A116540, A270995, A296119, A318360.
For normal multisets see A292432, A292444, A116539.
These partitions are ranked by A293243, complement A382200.
The MM-numbers of these multiset partitions (set of sets) are A302494.
Twice-partitions of this type are counted by A358914.
For distinct sums we have A381990 (ranks A381806), complement A381992 (ranks A382075).
The complement is counted by A382077, unique A382079.
A000041 counts integer partitions, strict A000009.
A050320 counts multiset partitions of prime indices into sets.
A050326 counts multiset partitions into distinct sets, complement A050345.
A265947 counts refinement-ordered pairs of integer partitions.
Cf. A293511, A299202, A317142, A381441, A381454, A381717, A381718, A381870.

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]]]];
Table[Length[Select[IntegerPartitions[n],Length[Select[mps[#],UnsameQ@@#&&And@@UnsameQ@@@#&]]==0&]],{n,0,9}]

a(19)-a(50) from Bert Dobbelaere, Mar 29 2025

A382216 Number of normal multisets of size n that can be partitioned into a set of sets with distinct sums.

Original entry on oeis.org

1, 1, 1, 3, 5, 11, 23, 48, 101, 208, 434

Offset: 0

Gus Wiseman, Mar 29 2025

We call a multiset normal iff it covers an initial interval of positive integers. The size of a multiset is the number of elements, counting multiplicity.

			The multiset {1,2,2,3,3} can be partitioned into a set of sets with distinct sums in 4 ways:
  {{2,3},{1,2,3}}
  {{2},{3},{1,2,3}}
  {{2},{1,3},{2,3}}
  {{1},{2},{3},{2,3}}
so is counted under a(5).
The multisets counted by A382214 but not by A382216 are:
  {1,1,1,1,2,2,3,3,3}
  {1,1,2,2,2,2,3,3,3}
The a(1) = 1 through a(5) = 11 multisets:
  {1}  {1,2}  {1,1,2}  {1,1,2,2}  {1,1,1,2,3}
              {1,2,2}  {1,1,2,3}  {1,1,2,2,3}
              {1,2,3}  {1,2,2,3}  {1,1,2,3,3}
                       {1,2,3,3}  {1,1,2,3,4}
                       {1,2,3,4}  {1,2,2,2,3}
                                  {1,2,2,3,3}
                                  {1,2,2,3,4}
                                  {1,2,3,3,3}
                                  {1,2,3,3,4}
                                  {1,2,3,4,4}
                                  {1,2,3,4,5}

Twice-partitions of this type are counted by A279785, without distinct sums A358914.
Factorizations of this type are counted by A381633, without distinct sums A050326.
Normal multiset partitions of this type are counted by A381718, A116539.
The complement is counted by A382202.
Without distinct sums we have A382214, complement A292432.
The case of a unique choice is counted by A382459, without distinct sums A382458.
For Heinz numbers: A293243, A381806, A382075, A382200.
For integer partitions: A381990, A381992, A382077, A382078.
Strong version: A382523, A382430, A381996, A292444.
Normal multiset partitions: A034691, A035310, A255906.
Set systems: A050342, A296120, A318361.
Set multipartitions: A089259, A270995, A296119, A318360.
Cf. A000110, A000670, A007716, A050320, A116540, A255903, A275780, A317532, A326518, A326519, A382429, A382460.

allnorm[n_]:=If[n<=0,{{}},Function[s,Array[Count[s,y_/;y<=#]+1&,n]]/@Subsets[Range[n-1]+1]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[mset_]:=Union[Sort[Sort/@(#/.x_Integer:>mset[[x]])]&/@sps[Range[Length[mset]]]];
Table[Length[Select[allnorm[n],Length[Select[mps[#],And@@UnsameQ@@@#&&UnsameQ@@Total/@#&]]>0&]],{n,0,5}]

A094000 Number of n X n (0,1)-matrices with no zero rows or columns and with all rows distinct and all columns distinct, up to permutation of rows.

Original entry on oeis.org

1, 1, 3, 29, 1015, 126651, 53354350, 74698954306, 350688201987402, 5624061753186933530, 314512139441575825493524, 62498777166571927258267336860, 44831219113504221199415663547412096

Offset: 0

Goran Kilibarda and Vladeta Jovovic, May 30 2004

Main diagonal of A059202.

G. Kilibarda and V. Jovovic, "Enumeration of some classes of T_0-hypergraphs", in

G. C. Greubel, Table of n, a(n) for n = 0..59
Goran Kilibarda and Vladeta Jovovic, Enumeration of some classes of T_0-hypergraphs, arXiv:1411.4187 [math.CO], 2014.

Cf. A048291, A059202, A088309.

Binary matrices with distinct rows and columns, various versions: A059202, A088309, A088310, A088616, A089673, A089674, A093466, A094000, A094223, A116532, A116539, A181230, A259763

f[n_] := Sum[ StirlingS1[n + 1, k] Binomial[2^(k - 1) - 1, n], {k, 0, n + 1}]; Table[ f[n], {n, 0, 12}] (* Robert G. Wilson v, Jun 01 2004 *)

a(n) = sum(k=0, n+1, stirling(n+1, k, 1)*binomial(2^(k-1)-1, n)); \\ Michel Marcus, Dec 17 2022

a(n) = Sum_{k=0..n+1} Stirling1(n+1, k)*binomial(2^(k-1)-1, n).

a(n) ~ binomial(2^n,n). - Vaclav Kotesovec, Mar 18 2014

More terms from Robert G. Wilson v, Jun 01 2004

A382200 Numbers that can be written as a product of distinct squarefree numbers.

Original entry on oeis.org

1, 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 26, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84

Offset: 1

Gus Wiseman, Mar 21 2025

nonn

First differs from A339741 in having 1080.
First differs from A382075 in having 18000.
These are positions of positive terms in A050326, complement A293243.
Also numbers whose prime indices can be partitioned into distinct sets.
Differs from A212167, which does not include 18000 = 2^4*3^2*5^3, for example. - R. J. Mathar, Mar 23 2025

			The prime indices of 1080 are {1,1,1,2,2,2,3}, and {{1},{2},{1,2},{1,2,3}} is a partition into a set of sets, so 1080 is in the sequence.
We have 18000 = 2*5*6*10*30, so 18000 is in the sequence.

Robert Israel, Table of n, a(n) for n = 1..10000

Twice-partitions of this type are counted by A279785, see also A358914.
Normal multisets not of this type are counted by A292432, strong A292444.
The complement is A293243, counted by A050342.
The case of a unique choice is A293511.
MM-numbers of multiset partitions into distinct sets are A302494.
For distinct block-sums instead of blocks we have A382075, counted by A381992.
Partitions of this type are counted by A382077, complement A382078.
Normal multisets of this type are counted by A382214, strong A381996.
A001055 counts multiset partitions of prime indices, strict A045778.
A050320 counts multiset partitions of prime indices into sets.
A050326 counts multiset partitions of prime indices into distinct sets.
A317141 counts coarsenings of prime indices, refinements A300383.
Cf. A000720, A001222, A005117, A050345, A089259, A116539, A270995, A381441, A382201, A382216.

N:= 1000: # to get all terms <= N
A:= Vector(N):
A[1]:= 1:
for n from 2 to N do
  if numtheory:-issqrfree(n) then
      S:= [$1..N/n]; T:= n*S; A[T]:= A[T]+A[S]
    fi;
od:
remove(t -> A[t]=0, [$1..N]); # Robert Israel, Apr 21 2025

sqfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfacs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Select[Range[100],Length[sqfacs[#]]>0&]

A088309 Number of equivalence classes of n X n (0,1)-matrices with all rows distinct and all columns distinct.

Original entry on oeis.org

1, 2, 5, 44, 1411, 159656, 62055868, 82060884560, 371036717493194, 5812014504668066528, 320454239459072905856944, 63156145369562679089674952768, 45090502574837184532027563736271152, 117910805393665959622047902193019284914432, 1139353529410754170844431642119963019965901238144

Offset: 0

N. J. A. Sloane, Nov 07 2003

nonn

Two such matrices are equivalent if they differ just by a permutation of the rows.

			a(2) = 5: 00/01, 00/10, 01/10, 01/11, 10/11.

G. C. Greubel, Table of n, a(n) for n = 0..59
G. Kilibarda and V. Jovovic, Enumeration of some classes of T_0-hypergraphs, arXiv:1411.4187 [math.CO], 2014.

Cf. A088229, A088310, A088616.
Main diagonal of A059084.
Binary matrices with distinct rows and columns, various versions: A059202, this sequence, A088310, A088616, A089673, A089674, A093466, A094000, A094223, A116532, A116539, A181230, A259763.

A088309:= func< n | (&+[Binomial(2^k,n)*StirlingFirst(n,k): k in [0..n]]) >;
[A088309(n): n in [0..30]]; // G. C. Greubel, Dec 15 2022

A088309[n_]:= A088309[n]=Sum[Binomial[2^j,n]*StirlingS1[n,j], {j,0,n}];
Table[A088309[n], {n,0,30}] (* G. C. Greubel, Dec 15 2022 *)

a(n) = sum(k=0, n, stirling(n, k, 1)*binomial(2^k, n)); \\ Michel Marcus, Dec 16 2022

@CachedFunction
def A088309(n): return (-1)^n*sum((-1)^k*binomial(2^k, n)*stirling_number1(n, k) for k in (0..n))
[A088309(n) for n in range(31)] # G. C. Greubel, Dec 15 2022

a(n) = Sum_{k=0..n} Stirling1(n, k)*binomial(2^k, n). - Vladeta Jovovic, Nov 07 2003

a(n) = A088310(n) / n!.

Suggested by Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 06 2003

a(0)-a(5) from W. Edwin Clark, Nov 07 2003

cp's OEIS Frontend

A382077 Number of integer partitions of n that can be partitioned into a set of sets.

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A181230 Square array T(m,n) giving the number of m X n (0,1)-matrices with pairwise distinct rows and pairwise distinct columns.

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A059202 Triangle T(n,m) of numbers of m-block T_0-covers of a labeled n-set, m = 0..2^n - 1.

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A088310 Number of n X n (0,1)-matrices with all rows distinct and all columns distinct.

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A116532 Number of singular n X n rational {0,1}-matrices with no zero rows or columns and with all rows distinct, up to permutation of rows.

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A382078 Number of integer partitions of n that cannot be partitioned into a set of sets.

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Mathematica

Extensions

A382216 Number of normal multisets of size n that can be partitioned into a set of sets with distinct sums.

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Mathematica

A094000 Number of n X n (0,1)-matrices with no zero rows or columns and with all rows distinct and all columns distinct, up to permutation of rows.

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Mathematica