A320423
Number of set partitions of {1,...,n} where each block's elements are pairwise coprime.
Original entry on oeis.org
1, 1, 1, 2, 2, 8, 4, 28, 18, 120, 60, 888, 252, 5220, 1860, 22224, 9552, 311088, 59616, 2473056, 565920, 13627008, 4051872, 235039392, 33805440, 1932037632, 465239808, 20604487680, 4294865664, 386228795904, 35413136640
Offset: 0
The a(5) = 8 set partitions:
{{1},{2,3},{4,5}}
{{1},{2,5},{3,4}}
{{1,2},{3,4,5}}
{{1,4},{2,3,5}}
{{1,2,3},{4,5}}
{{1,2,5},{3,4}}
{{1,3,4},{2,5}}
{{1,4,5},{2,3}}
Cf.
A000110,
A051424,
A084422,
A085945,
A186974,
A187106,
A302569,
A302696,
A303139,
A303140,
A320424,
A320426,
A320430,
A320768.
-
spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@spsu[Select[foo,Complement[#,Complement[set,s]]=={}&],Complement[set,s]]]/@Cases[foo,{i,_}];
Table[Length[spsu[Select[Subsets[Range[n]],CoprimeQ@@#&],Range[n]]],{n,10}]
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
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
Showing 1-2 of 2 results.
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