A059849 -id:A059849 - cp's OEIS frontend

A060639 Number of pairs of partitions of [n] whose join is the partition {{1,2,...,n}}.

1, 1, 3, 15, 119, 1343, 19905, 369113, 8285261, 219627683, 6746244739, 236561380795, 9356173080985, 413251604702069, 20215438754502217, 1087524296159855603, 63950948621703499839, 4089003767746536828183, 282970817307108139386841, 21107742616278461624923449, 1690957890908364634072451893

Offset: 0

E. R. Canfield (erc(AT)cs.uga.edu), Apr 16 2001

nonn

It appears that a(n) = 2*A001188(n) - 1 for n > 0. This holds for the first 50 terms. - Charles R Greathouse IV, Mar 21 2012

			J(2) = 3 because there are two partitions of {1,2} and of the four pairs of partitions, only the pair ( {{1},{2}}, {{1},{2}} ) fails to have join {{1,2}}.

Vincenzo Librandi, Table of n, a(n) for n = 0..325
E. R. Canfield, Meet and join in the partition lattice, Electronic Journal of Combinatorics, 8 (2001) R15.
I. Dolinka, J. East, A. Evangelou, D. FitzGerald, N. Ham, et al., Enumeration of idempotents in diagram semigroups and algebras, arXiv preprint arXiv:1408.2021 [math.GR], 2014; Table 3.
B. Pittel, Where the typical set partitions meet and join, Electronic Journal of Combinatorics, 7 (2000) R5.
Frank Simon, Algebraic Methods for Computing the Reliability of Networks, Dissertation, Doctor Rerum Naturalium (Dr. rer. nat.), Fakultät Mathematik und Naturwissenschaften der Technischen Universität Dresden, 2012.

Bell numbers: A000110, Stirling numbers of the second kind: A000225, number of pairs whose meet equals {{1}, {2}, ..., {n}}: A059849.

Cf. A001188, A318815.

list[n_] := Module[{t}, t = Table[BellB[k-1]^2/(k-1)!, {k, 1, n+1}]; CoefficientList[1+Log[O[x]^(n+1)+Sum[t[[i]] x^(i-1), {i, 1, Length[t]}]], x]];
list[17] Range[0, 17]! (* Jean-François Alcover, Nov 03 2018, from PARI *)

Bell(n)=round(suminf(k=0,k^n/k!)/exp(1))
list(n)=my(v=Vec(log(O(x^(n+1))+Polrev(vector(n+1,k,Bell(k-1)^2/(k-1)!)))));concat(1,vector(n,i,v[i]*i!)) \\ Charles R Greathouse IV, Mar 21 2012

The e.g.f. J(x) satisfies the equation Sum_{n>=0} (B_n)^2 x^n/n! = exp(J(x)-1), where B_n is the n-th Bell number.

a(0) = 1; a(n) = Bell(n)^2 - (1/n) * Sum_{k=1..n-1} binomial(n,k) * Bell(n-k)^2 * k * a(k). - Ilya Gutkovskiy, Jan 17 2020

More terms from Vladeta Jovovic, Apr 18 2001

A318393 Regular tetrangle where T(n,k,i) is the number of pairs of set partitions of {1,...,n} with meet of length k and join of length i.

Original entry on oeis.org

1, 1, 2, 1, 1, 6, 3, 8, 6, 1, 1, 14, 7, 48, 36, 6, 56, 44, 12, 1, 1, 30, 15, 200, 150, 25, 560, 440, 120, 10, 552, 440, 140, 20, 1, 1, 62, 31, 720, 540, 90, 3640, 2860, 780, 65, 8280, 6600, 2100, 300, 15, 7202, 5632, 1920, 340, 30, 1, 1, 126, 63, 2408, 1806

Offset: 1

Gus Wiseman, Aug 25 2018

			The T(3,3,1) = 8 pairs of set partitions:
  {{1},{2},{3}}  {{1,2,3}}
   {{1},{2,3}}  {{1,2},{3}}
   {{1},{2,3}}  {{1,3},{2}}
   {{1,2},{3}}  {{1},{2,3}}
   {{1,2},{3}}  {{1,3},{2}}
   {{1,3},{2}}  {{1},{2,3}}
   {{1,3},{2}}  {{1,2},{3}}
    {{1,2,3}}  {{1},{2},{3}}
Tetrangle begins:
   1   1     1       1            1
       2 1   6 3     14 7         30  15
             8 6 1   48 36 6      200 150 25
                     56 44 12 1   560 440 120 10
                                  552 440 140 20  1

Cf. A000110, A000258, A001247, A008277, A048994, A059849, A060639, A181939, A318389, A318390, A318391, A318392.

csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
spmeet[a_,b_]:=DeleteCases[Union@@Outer[Intersection,a,b,1],{}];spmeet[a_,b_,c__]:=spmeet[spmeet[a,b],c];
Table[Length[Select[Tuples[sps[Range[n]],2],And[Length[spmeet@@#]==k,Length[csm[Union@@#]]==j]&]],{n,6},{k,n},{j,k}]

A181939 Number of pairs of set partitions of {1,2,...,n} whose meet is {{1},{2},...,{n}} and join is {{1,2,...,n}}.

Original entry on oeis.org

1, 1, 2, 8, 56, 552, 7202, 118456, 2369922, 56230544, 1552048082, 49080888144, 1756527398738, 70427165428648, 3136819046716266, 154090456510590632, 8296738497931578818, 487014208107376581984, 31018372994440588508642, 2134584265273475942046304

Offset: 0

Alexander Steinhardt (asteinh1(AT)hs-mittweida.de), Jens Schreiter (jschrei1(AT)hs-mittweida.de), Frank Simon, Apr 03 2012

			For n = 2 there are exactly the following two pairs ({{1,2}},{{1},{2}}), ({{1},{2}},{{1,2}}) satisfying the imposed conditions.

Alois P. Heinz, Table of n, a(n) for n = 0..325
E. R. Canfield, Meet and join in the partition lattice, Electronic Journal of Combinatorics, 8 (2001) R15.
B. Pittel, Where the typical set partitions meet and join, Electronic Journal of Combinatorics, 7 (2000) R5.
Frank Simon, Algebraic Methods for Computing the Reliability of Networks, Dissertation, Doctor Rerum Naturalium (Dr. rer. nat.), Fakultät Mathematik und Naturwissenschaften der Technischen Universität Dresden, 2012. See Table 3.3. - _N. J. A. Sloane_, Jan 04 2013

Cf. A059849, A060639.

with(combinat):
m:= proc(n) option remember; add(stirling1(n, k)*bell(k)^2, k=0..n) end:
a:= proc(n) option remember;
      m(n) -add(binomial(n-1,k)*m(k)*a(n-k), k=1..n-1)
    end:
seq(a(n), n=0..20); # Alois P. Heinz, Apr 20 2012

m[n_] := m[n] = Sum[StirlingS1[n, k]*BellB[k]^2, {k, 0, n}]; a[n_] := a[n] = m[n] - Sum[ Binomial[n-1, k]*m[k]*a[n-k], {k, 1, n-1}]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 15 2015, after Alois P. Heinz *)

E.g.f.: 1+log(M(x)), where M(x) is the e.g.f. of A059849 of all pairs of set partitions of {1,2,...,n} whose meet is {{1},{2},...,{n}}.

a(n) = m(n) - Sum_{k=1..n-1} C(n-1,k)*m(k)*a(n-k), where m(n) = A059849(n) of all pairs of set partitions of an n-element set having meet {{1},{2},...,{n}}.

Terms corrected and more terms added, Alois P. Heinz, Apr 20 2012

A318391 Regular triangle where T(n,k) is the number of pairs of set partitions of {1,...,n} with meet of length k.

Original entry on oeis.org

1, 1, 3, 1, 9, 15, 1, 21, 90, 113, 1, 45, 375, 1130, 1153, 1, 93, 1350, 7345, 17295, 15125, 1, 189, 4515, 39550, 161420, 317625, 245829, 1, 381, 14490, 192213, 1210650, 4023250, 6883212, 4815403, 1, 765, 45375, 878010, 8014503, 40020750, 113572998, 173354508, 111308699

Offset: 1

Gus Wiseman, Aug 25 2018

			The T(3,2) = 9 pairs of set partitions:
  {{1},{2,3}}  {{1},{2,3}}
  {{1},{2,3}}   {{1,2,3}}
  {{1,2},{3}}  {{1,2},{3}}
  {{1,2},{3}}   {{1,2,3}}
  {{1,3},{2}}  {{1,3},{2}}
  {{1,3},{2}}   {{1,2,3}}
   {{1,2,3}}   {{1},{2,3}}
   {{1,2,3}}   {{1,2},{3}}
   {{1,2,3}}   {{1,3},{2}}
Triangle begins:
     1
     1     3
     1     9    15
     1    21    90   113
     1    45   375  1130  1153
     1    93  1350  7345 17295 15125

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Row sums are A001247. Last column is A059849.

Cf. A000110, A000258, A008277, A048994, A060639, A181939, A318389, A318390, A318392, A318393.

Table[StirlingS2[n,k]*Sum[StirlingS1[k,i]*BellB[i]^2,{i,k}],{n,10},{k,n}]

row(n) = {my(b=Vec(serlaplace(exp(exp(x + O(x*x^n))-1)-1))); vector(n, k, stirling(n,k,2)*sum(i=1, k, stirling(k,i,1)*b[i]^2))}
{ for(n=1, 10, print(row(n))) } \\ Andrew Howroyd, Jan 19 2023

T(n,k) = S(n,k) * Sum_{i=1..k} s(k,i) * B(i)^2 where S = A008277, s = A048994, B = A000110.

A318389 Regular triangle where T(n,k) is the number of pairs of set partitions of {1,...,n} with meet {{1},...,{n}} and join of length k.

Original entry on oeis.org

1, 2, 1, 8, 6, 1, 56, 44, 12, 1, 552, 440, 140, 20, 1, 7202, 5632, 1920, 340, 30, 1, 118456, 89278, 31192, 6160, 700, 42, 1, 2369922, 1708016, 595448, 124432, 16240, 1288, 56, 1, 56230544, 38592786, 13214672, 2830632, 400512, 37296, 2184, 72, 1, 1552048082

Offset: 1

Gus Wiseman, Aug 25 2018

			The T(3,2) = 6 pairs of set partitions:
  {{1},{2},{3}}  {{1},{2,3}}
  {{1},{2},{3}}  {{1,2},{3}}
  {{1},{2},{3}}  {{1,3},{2}}
   {{1},{2,3}}  {{1},{2},{3}}
   {{1,2},{3}}  {{1},{2},{3}}
   {{1,3},{2}}  {{1},{2},{3}}
Triangle begins:
     1
     2    1
     8    6    1
    56   44   12    1
   552  440  140   20    1
  7202 5632 1920  340   30    1

Row sums are A059849. First column is A181939.

Cf. A000110, A000258, A001247, A008277, A048994, A060639, A318390, A318391, A318392, A318393.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
spmeet[a_,b_]:=DeleteCases[Union@@Outer[Intersection,a,b,1],{}];spmeet[a_,b_,c__]:=spmeet[spmeet[a,b],c];
Table[Length[Select[Tuples[sps[Range[n]],2],And[Max@@Length/@spmeet@@#==1,Length[csm[Union@@#]]==k]&]],{n,5},{k,n}]

A318396 Number of pairs of integer partitions (y, v) of n such that there exists a pair of set partitions of {1,...,n} with meet {{1},...,{n}}, the first having block sizes y and the second v.

Original entry on oeis.org

1, 1, 3, 6, 15, 28, 64, 116, 238, 430, 818, 1426, 2618, 4439, 7775, 12993, 22025, 35946, 59507, 95319, 154073, 243226, 385192, 598531, 933096, 1429794, 2193699, 3322171, 5027995, 7524245, 11253557, 16661211, 24637859, 36130242, 52879638, 76830503, 111422013, 160505622

Offset: 0

Gus Wiseman, Aug 25 2018

nonn

A multiset is normal if it spans an initial interval of positive integers, and strongly normal if in addition it has weakly decreasing multiplicities. a(n) is also the number of combinatory separations (see A269134 for definition) of strongly normal multisets of size n into normal sets.
From Andrew Howroyd, Oct 31 2019: (Start)
Also, the number of distinct unordered row and column sums of binary matrices without empty columns or rows and with a total of n ones. Only matrices in which both row and columns sums are weakly increasing need to be considered.
By the Gale-Ryser theorem this is equivalent to the number of pairs of integer partitions (y,v) of n with y dominating v. (End)

			The a(4) = 15 pairs of integer partitions:
     4, 1111
    22, 22
    22, 211
    22, 1111
    31, 211
    31, 1111
   211, 22
   211, 31
   211, 211
   211, 1111
  1111, 4
  1111, 22
  1111, 31
  1111, 211
  1111, 1111
The a(4) = 15 combinatory separations:
  1111<={1,1,1,1}
  1112<={1,1,12}
  1112<={1,1,1,1}
  1122<={12,12}
  1122<={1,1,12}
  1122<={1,1,1,1}
  1123<={1,123}
  1123<={12,12}
  1123<={1,1,12}
  1123<={1,1,1,1}
  1234<={1234}
  1234<={1,123}
  1234<={12,12}
  1234<={1,1,12}
  1234<={1,1,1,1}

Andrew Howroyd, Table of n, a(n) for n = 0..100
Manfred Krause, A simple proof of the Gale-Ryser theorem, American Mathematical Monthly, 1996.
Wikipedia, Gale-Ryser theorem

Cf. A000041, A000110, A001247, A007716, A008277, A029894, A049311, A059849, A116540, A181939, A265947, A269134, A318393, A318394, A327913.

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]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
normize[m_]:=m/.Rule@@@Table[{Union[m][[i]],i},{i,Length[Union[m]]}];
Table[Length[Select[Union@@Table[{m,Sort[normize/@#]}&/@mps[m],{m,strnorm[n]}],And@@UnsameQ@@@#[[2]]&]],{n,6}]

IsDom(p,q)=if(#q<#p, 0, my(s=0,t=0); for(i=0, #p-1, s+=p[#p-i]; t+=q[#q-i]; if(t>s, return(0))); 1)
a(n)={if(n<1, n==0, my(s=0); forpart(p=n, forpart(q=n, s+=IsDom(p,q), [1, p[#p]], [#p, n])); s)} \\ Andrew Howroyd, Oct 31 2019

\\ faster version.
a(n)={local(Cache=Map());
  my(recurse(b, c, s, t)=my(hk=Vecsmall([b, c, s, t]), z);
     if(!mapisdefined(Cache, hk, &z),
       z = if(s, sum(i=1, min(s, b), sum(j=1, min(t-s+i, c), self()(i, j, s-i, t-j))),
           if(t, sum(j=1, min(t, c), self()(b, j, s, t-j)), 1));
       mapput(Cache, hk, z)); z);
  recurse(n, n, n, n)
} \\ Andrew Howroyd, Oct 31 2019

Terms a(9) and beyond from Andrew Howroyd, Oct 31 2019

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

A318392 Regular triangle where T(n,k) is the number of pairs of set partitions of {1,...,n} with join of length k.

Original entry on oeis.org

1, 3, 1, 15, 9, 1, 119, 87, 18, 1, 1343, 1045, 285, 30, 1, 19905, 15663, 4890, 705, 45, 1, 369113, 286419, 95613, 16450, 1470, 63, 1, 8285261, 6248679, 2147922, 410053, 44870, 2730, 84, 1, 219627683, 159648795, 55211229, 11202534, 1394883, 105714, 4662, 108, 1

Offset: 1

Gus Wiseman, Aug 25 2018

			The T(3,2) = 9 pairs of set partitions:
  {{1},{2},{3}}  {{1},{2,3}}
  {{1},{2},{3}}  {{1,2},{3}}
  {{1},{2},{3}}  {{1,3},{2}}
   {{1},{2,3}}  {{1},{2},{3}}
   {{1},{2,3}}   {{1},{2,3}}
   {{1,2},{3}}  {{1},{2},{3}}
   {{1,2},{3}}   {{1,2},{3}}
   {{1,3},{2}}  {{1},{2},{3}}
   {{1,3},{2}}   {{1,3},{2}}
Triangle begins:
      1
      3     1
     15     9     1
    119    87    18     1
   1343  1045   285    30     1
  19905 15663  4890   705    45     1

Row sums are A001247. First column is A060639.

Cf. A000110, A000258, A008277, A048994, A059849, A181939, A318389, A318390, A318391, A318393.

nn=5;Table[n!*SeriesCoefficient[Sum[BellB[n]^2*x^n/n!,{n,0,nn}]^t,{x,0,n},{t,0,k}],{n,nn},{k,n}]

E.g.f.: (Sum_{n>=0} B(n)^2 x^n/n!)^t where B = A000110.

A322441 Number of pairs of set partitions of {1,...,n} where no block of one is a subset or equal to any block of the other.

Original entry on oeis.org

1, 0, 0, 0, 6, 60, 630, 9660, 192906

Offset: 0

Gus Wiseman, Dec 08 2018

For any pair (X,Y) meeting the requirement, so does the pair (Y,X) which must be distinct from (X,Y), except for X = Y = {} when n = 0. Therefore all a(n) are even for n > 0. - M. F. Hasler, Dec 30 2020

			The a(4) = 6 pairs of set partitions:
  {{1,2},{3,4}} and {{1,3},{2,4}},
  {{1,2},{3,4}} and {{1,4},{2,3}},
  {{1,3},{2,4}} and {{1,2},{3,4}},
  {{1,3},{2,4}} and {{1,4},{2,3}},
  {{1,4},{2,3}} and {{1,2},{3,4}},
  {{1,4},{2,3}} and {{1,3},{2,4}}.

Cf. A000110, A000258, A001247, A008277, A059849, A060639, A181939, A318393, A321760 (unlabeled version), A322435, A322442.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
stabQ[u_]:=stabQ[u,SubsetQ];stabQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[Tuples[sps[Range[n]],2],And[UnsameQ@@Join@@#,stabQ[Join@@#]]&]],{n,6}]

A318390 Regular triangle where T(n,k) is the number of pairs of set partitions of {1,...,n} with join {{1,...,n}} and meet of length k.

Original entry on oeis.org

1, 1, 2, 1, 6, 8, 1, 14, 48, 56, 1, 30, 200, 560, 552, 1, 62, 720, 3640, 8280, 7202, 1, 126, 2408, 19600, 77280, 151242, 118456, 1, 254, 7728, 95256, 579600, 1915732, 3316768, 2369922, 1, 510, 24200, 435120, 3836952, 19056492, 54726672, 85317192, 56230544, 1

Offset: 1

Gus Wiseman, Aug 25 2018

			The T(3,3) = 8 pairs of set partitions:
  {{1},{2},{3}}  {{1,2,3}}
   {{1},{2,3}}  {{1,2},{3}}
   {{1},{2,3}}  {{1,3},{2}}
   {{1,2},{3}}  {{1},{2,3}}
   {{1,2},{3}}  {{1,3},{2}}
   {{1,3},{2}}  {{1},{2,3}}
   {{1,3},{2}}  {{1,2},{3}}
    {{1,2,3}}  {{1},{2},{3}}
Triangle begins:
    1
    1    2
    1    6    8
    1   14   48   56
    1   30  200  560  552
    1   62  720 3640 8280 7202

Row sums are A060639. Last column is A181939.

Cf. A000110, A000258, A001247, A008277, A048994, A059849, A318389, A318391, A318392, A318393.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
csm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[OrderedQ[#],UnsameQ@@#,Length[Intersection@@s[[#]]]>0]&]},If[c=={},s,csm[Union[Append[Delete[s,List/@c[[1]]],Union@@s[[c[[1]]]]]]]]];
spmeet[a_,b_]:=DeleteCases[Union@@Outer[Intersection,a,b,1],{}];spmeet[a_,b_,c__]:=spmeet[spmeet[a,b],c];
Table[Length[Select[Tuples[sps[Range[n]],2],And[Length[spmeet@@#]==k,Length[csm[Union@@#]]==1]&]],{n,6},{k,n}]

T(n,k) = S(n,k) * A181939(k) where S = A008277.

cp's OEIS Frontend

A060639 Number of pairs of partitions of [n] whose join is the partition {{1,2,...,n}}.

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A318393 Regular tetrangle where T(n,k,i) is the number of pairs of set partitions of {1,...,n} with meet of length k and join of length i.

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A181939 Number of pairs of set partitions of {1,2,...,n} whose meet is {{1},{2},...,{n}} and join is {{1,2,...,n}}.

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Maple

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A318391 Regular triangle where T(n,k) is the number of pairs of set partitions of {1,...,n} with meet of length k.

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A318389 Regular triangle where T(n,k) is the number of pairs of set partitions of {1,...,n} with meet {{1},...,{n}} and join of length k.

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A318396 Number of pairs of integer partitions (y, v) of n such that there exists a pair of set partitions of {1,...,n} with meet {{1},...,{n}}, the first having block sizes y and the second v.

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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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A318392 Regular triangle where T(n,k) is the number of pairs of set partitions of {1,...,n} with join of length k.

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