A060639 -id:A060639 - cp's OEIS frontend

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.

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}]

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.

A256033 Number of idempotents of rank 1 in partition monoid P_n.

Original entry on oeis.org

1, 5, 43, 529, 8451, 167397, 3984807, 111319257, 3583777723, 131082199809, 5385265586075, 246172834737485, 12422776100542887, 687441750763500441, 41475644663003037947, 2714680813135603845921

Offset: 1

N. J. A. Sloane, Mar 14 2015

nonn

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. See Table 3.

Cf. A060639, A256034.

e256033 := proc(n,r,s)
    option remember;
    local resu,m,a,b;
    if n <= 0 then
        return 0;
    end if;
    if s = 1 then
        combinat[stirling2](n,r) ;
    elif r= 1 then
        combinat[stirling2](n,s) ;
    else
        resu := s*procname(n-1,r-1,s)+r*procname(n-1,r,s-1)+r*s*procname(n-1,r,s) ;
        for m from 1 to n-2 do
        for a from 1 to r-1 do
        for b from 1 to s-1 do
            resu := resu + binomial(n-2,m) *(a*(s-b)+b*(r-a))
                *procname(m,a,b)*procname(n-m-1,r-a,s-b);
        end do:
        end do:
        end do:
        resu ;
    end if;
end proc:
A256033 := proc(n)
    a := 0 ;
    for r from 1 to n do
    for s from 1 to n do
        a  := a+r*s*e256033(n,r,s) ;
    end do;
    end do;
end proc:
seq(A256033(n),n=1..16) ; # R. J. Mathar, Mar 23 2015

f[n_, r_, s_] := f[n, r, s] = Module[{resu, m, a, b}, Which[n <= 0, 0, s == 1, StirlingS2[n, r], r == 1, StirlingS2[n, s], True, resu = s*f[n - 1, r - 1, s] + r*f[n - 1, r, s - 1] + r*s*f[n - 1, r, s]; Do[resu += Binomial[n - 2, m]*(b*(r - a) + a*(s - b))*f[m, a, b]*f[-m + n - 1, r - a, s - b], {m, n}, {a, r - 1}, {b, s - 1}]; resu]];
a[n_] := Module[{b = 0}, Do[b += r*s*f[n, r, s], {r, n}, {s, n}]; b];
Array[a, 16] (* Jean-François Alcover, Nov 23 2017, after R. J. Mathar *)

@cached_function
def F(n, r, s):
    if n <= 0: return 0
    if s == 1: return stirling_number2(n, r)
    if r == 1: return stirling_number2(n, s)
    ret = s*F(n-1,r-1,s)+r*F(n-1,r,s-1)+r*s*F(n-1,r,s)
    for m in (1..n-2):
        for a in (1..r-1):
            for b in (1..s-1):
                ret += binomial(n-2,m)*(a*(s-b)+b*(r-a))*F(m,a,b)*F(n-m-1,r-a,s-b)
    return ret
@cached_function
def A256033(n):
    a = 0
    for r in (1..n):
        for s in (1..n):
            a += r*s*F(n, r, s)
    return a
[A256033(n) for n in (1..9)] # Peter Luschny, Jan 17 2016

A318394 Number of finite sets of set partitions of {1,...,n} such that any two have meet {{1},...,{n}}.

Original entry on oeis.org

2, 4, 18, 316, 37492

Offset: 1

Gus Wiseman, Aug 25 2018

			The a(3) = 18 sets of set partitions:
        0
    {{1,2,3}}
   {{1,3},{2}}
   {{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},{2},{3}}  {{1,3},{2}}
  {{1},{2},{3}}  {{1,2},{3}}
  {{1},{2},{3}}  {{1},{2,3}}
   {{1},{2,3}}   {{1,2},{3}}  {{1,3},{2}}
  {{1},{2},{3}}  {{1,2},{3}}  {{1,3},{2}}
  {{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}}  {{1,3},{2}}

Cf. A000110, A000258, A008277, A059849, A060639, A096827, A181939.

Cf. A318389, A318390, A318391, A318392, A318531, A318532, A318531, A318532.

stableSets[u_,Q_]:=If[Length[u]===0,{{}},With[{w=First[u]},Join[stableSets[DeleteCases[u,w],Q],Prepend[#,w]&/@stableSets[DeleteCases[u,r_/;r===w||Q[r,w]||Q[w,r]],Q]]]];
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[stableSets[sps[Range[n]],Max@@Length/@spmeet[#1,#2]>1&]],{n,5}]

cp's OEIS Frontend

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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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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Mathematica

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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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.

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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.

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A256033 Number of idempotents of rank 1 in partition monoid P_n.