A305194 Number of z-forests summing to n. Number of strict integer partitions of n with pairwise indivisible parts and all connected components having clutter density -1.

1, 1, 1, 1, 2, 1, 3, 2, 3, 4, 5, 4, 6, 7, 7, 9, 11, 12, 13, 15, 17, 20, 23, 25, 27, 32, 35, 40, 45, 49, 54, 58, 67, 78, 82, 95, 99, 111, 123, 135, 150, 164, 177, 194, 214, 236, 260, 282, 309, 330

Offset

1,5

Comments

Given a finite set S of positive integers greater than 1, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices that have a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. A set S is said to be connected if G(S) is a connected graph. The clutter density of S is defined to be Sum_{s in S} (omega(s) - 1) - omega(lcm(S)), where omega = A001221 and lcm is least common multiple. Then a z-forest is a strict integer partition with pairwise indivisible parts greater than 1 such that all connected components have clutter density -1.

Links

Table of n, a(n) for n=1..50.

R. Bacher, On the enumeration of labelled hypertrees and of labelled bipartite trees, arXiv:1102.2708 [math.CO].

Example

The a(17) = 11 z-forests together with the corresponding multiset systems:
       (17): {{7}}
     (15,2): {{2,3},{1}}
     (14,3): {{1,4},{2}}
     (13,4): {{6},{1,1}}
     (12,5): {{1,1,2},{3}}
     (11,6): {{5},{1,2}}
     (10,7): {{1,3},{4}}
      (9,8): {{2,2},{1,1,1}}
   (10,4,3): {{1,3},{1,1},{2}}
    (7,6,4): {{4},{1,2},{1,1}}
  (7,5,3,2): {{4},{3},{2},{1}}

Mathematica

zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]], 2], And[Less@@#, GCD@@s[[#]]]>1&]}, If[c=={}, s, zsm[Union[Append[Delete[s, List/@c[[1]]], LCM@@s[[c[[1]]]]]]]]];
zensity[s_]:=Total[(PrimeNu[#]-1&)/@s]-PrimeNu[LCM@@s];
zreeQ[s_]:=And[Length[s]>=2, zensity[s]==-1];
Table[Length[Select[IntegerPartitions[n], Function[s, UnsameQ@@s&&And@@(Length[#]==1||zreeQ[#]&)/@Table[Select[s, Divisible[m, #]&], {m, zsm[s]}]&&Select[Tuples[s, 2], UnsameQ@@#&&Divisible@@#&]=={}]]], {n, 50}]

Crossrefs

Cf. A030019, A048143, A134954, A275307, A285572, A293510, A293993, A293994, A303362, A303837, A303838, A304118, A304382, A305078, A305195.

Sequence in context: A363708 A285779 A319320 * A303362 A362061 A183163

Adjacent sequences: A305191 A305192 A305193 * A305195 A305196 A305197

Keyword

nonn

Author

Gus Wiseman, May 27 2018

Status

approved