Displaying 1-10 of 19 results found.
Number of lone-child-avoiding locally disjoint rooted identity trees whose leaves form an integer partition of n.
+10
16
1, 1, 2, 3, 6, 13, 28, 62, 143, 338, 804, 1948, 4789, 11886, 29796, 75316, 191702, 491040, 1264926, 3274594, 8514784, 22229481, 58243870
COMMENTS
A rooted tree is lone-child-avoiding if every non-leaf node has at least two branches. It is locally disjoint if no branch overlaps any other (unequal) branch of the same root. It is an identity tree if no branch appears multiple times under the same root.
EXAMPLE
The a(7) = 28 rooted trees:
7,
(16),
(25),
(1(15)),
(34),
(1(24)), (2(14)), (4(12)), (124),
(1(1(14))),
(3(13)),
(2(23)),
(1(1(23))), (1(2(13))), (1(3(12))), (1(123)), (2(1(13))), (3(1(12))), (12(13)), (13(12)),
(1(1(1(13)))),
(2(2(12))),
(1(1(2(12)))), (1(2(1(12)))), (1(12(12))), (2(1(1(12)))), (12(1(12))),
(1(1(1(1(12))))).
Missing from this list but counted by A300660 are ((12)(13)) and ((12)(1(12))).
MATHEMATICA
disjointQ[u_]:=Apply[And, Outer[#1==#2||Intersection[#1, #2]=={}&, u, u, 1], {0, 1}];
nms[n_]:=nms[n]=Prepend[Join@@Table[Select[Union[Sort/@Tuples[nms/@ptn]], And[UnsameQ@@#, disjointQ[#]]&], {ptn, Rest[IntegerPartitions[n]]}], {n}];
Table[Length[nms[n]], {n, 10}]
CROSSREFS
The semi-identity tree version is A212804. Not requiring local disjointness gives A300660. The non-identity tree version is A316696. This is the case of A331686 where all leaves are singletons. Locally disjoint rooted identity trees are A316471. Lone-child-avoiding locally disjoint rooted trees are A331680. Locally disjoint enriched identity p-trees are A331684.
EXTENSIONS
Updated with corrected terminology by Gus Wiseman, Feb 06 2020
Number of lone-child-avoiding locally disjoint rooted identity trees whose leaves are integer partitions whose multiset union is an integer partition of n.
+10
15
1, 2, 4, 8, 17, 41, 103, 280, 793, 2330, 6979, 21291
COMMENTS
A rooted tree is locally disjoint if no child of any vertex has branches overlapping the branches of any other (unequal) child of the same vertex. Lone-child-avoiding means there are no unary branchings. In an identity tree, all branches of any given vertex are distinct.
EXAMPLE
The a(1) = 1 through a(5) = 17 trees:
(1) (2) (3) (4) (5)
(11) (12) (13) (14)
(111) (22) (23)
((1)(2)) (112) (113)
(1111) (122)
((1)(3)) (1112)
((2)(11)) (11111)
((1)((1)(2))) ((1)(4))
((2)(3))
((1)(22))
((3)(11))
((2)(111))
((1)((1)(3)))
((2)((1)(2)))
((11)((1)(2)))
((1)((2)(11)))
((1)((1)((1)(2))))
MATHEMATICA
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]]]];
disjointQ[u_]:=Apply[And, Outer[#1==#2||Intersection[#1, #2]=={}&, u, u, 1], {0, 1}];
mpti[m_]:=Prepend[Join@@Table[Select[Union[Sort/@Tuples[mpti/@p]], UnsameQ@@#&&disjointQ[#]&], {p, Select[mps[m], Length[#]>1&]}], m];
Table[Sum[Length[mpti[m]], {m, Sort/@IntegerPartitions[n]}], {n, 8}]
CROSSREFS
The non-identity version is A331678. The case where the leaves are all singletons is A316694. Locally disjoint identity trees are A316471. Locally disjoint enriched identity p-trees are A331684. Lone-child-avoiding locally disjoint rooted semi-identity trees are A212804. Cf. A000669, A001678, A005804, A141268, A300660, A316697, A319312, A331679, A331683, A331783, A331874, A331875.
Matula-Goebel numbers of lone-child-avoiding rooted semi-identity trees.
+10
15
1, 4, 8, 14, 16, 28, 32, 38, 56, 64, 76, 86, 106, 112, 128, 133, 152, 172, 212, 214, 224, 256, 262, 266, 301, 304, 326, 344, 371, 424, 428, 448, 512, 524, 526, 532, 602, 608, 622, 652, 688, 742, 749, 766, 817, 848, 856, 886, 896, 917, 1007, 1024, 1048, 1052
COMMENTS
First differs from A331683 in having 133, the Matula-Goebel number of the tree ((oo)(ooo)). Lone-child-avoiding means there are no unary branchings.
In a semi-identity tree, the non-leaf branches of any given vertex are all distinct.
The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.
Consists of one, and all composite numbers that are n times a power of two, where n is a squarefree number whose prime indices already belong to the sequence, and a prime index of n is a number m such that prime(m) divides n. [Clarified by Peter Munn and Gus Wiseman, Jun 24 2021]
EXAMPLE
The sequence of all lone-child-avoiding rooted semi-identity trees together with their Matula-Goebel numbers begins:
1: o
4: (oo)
8: (ooo)
14: (o(oo))
16: (oooo)
28: (oo(oo))
32: (ooooo)
38: (o(ooo))
56: (ooo(oo))
64: (oooooo)
76: (oo(ooo))
86: (o(o(oo)))
106: (o(oooo))
112: (oooo(oo))
128: (ooooooo)
133: ((oo)(ooo))
152: (ooo(ooo))
172: (oo(o(oo)))
212: (oo(oooo))
214: (o(oo(oo)))
The sequence of terms together with their prime indices begins:
1: {} 224: {1,1,1,1,1,4}
4: {1,1} 256: {1,1,1,1,1,1,1,1}
8: {1,1,1} 262: {1,32}
14: {1,4} 266: {1,4,8}
16: {1,1,1,1} 301: {4,14}
28: {1,1,4} 304: {1,1,1,1,8}
32: {1,1,1,1,1} 326: {1,38}
38: {1,8} 344: {1,1,1,14}
56: {1,1,1,4} 371: {4,16}
64: {1,1,1,1,1,1} 424: {1,1,1,16}
76: {1,1,8} 428: {1,1,28}
86: {1,14} 448: {1,1,1,1,1,1,4}
106: {1,16} 512: {1,1,1,1,1,1,1,1,1}
112: {1,1,1,1,4} 524: {1,1,32}
128: {1,1,1,1,1,1,1} 526: {1,56}
133: {4,8} 532: {1,1,4,8}
152: {1,1,1,8} 602: {1,4,14}
172: {1,1,14} 608: {1,1,1,1,1,8}
212: {1,1,16} 622: {1,64}
214: {1,28} 652: {1,1,38}
MATHEMATICA
csiQ[n_]:=n==1||!PrimeQ[n]&&FreeQ[FactorInteger[n], {_?(#>2&), _?(#>1&)}]&&And@@csiQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[100], csiQ]
CROSSREFS
The non-semi case is {1}.
Not requiring lone-child-avoidance gives A306202. The locally disjoint version is A331683. These trees are counted by A331966. The semi-lone-child-avoiding case is A331994. Matula-Goebel numbers of rooted identity trees are A276625. Matula-Goebel numbers of lone-child-avoiding rooted trees are A291636. Semi-identity trees are counted by A306200. Cf. A001678, A004111, A007097, A061775, A122132, A196050, A300660, A316694, A320269, A331681, A331686, A331875, A331936, A331937, A331993.
Matula-Goebel numbers of semi-lone-child-avoiding locally disjoint rooted trees.
+10
14
1, 2, 4, 6, 8, 9, 12, 14, 16, 18, 24, 26, 27, 28, 32, 36, 38, 46, 48, 49, 52, 54, 56, 64, 69, 72, 74, 76, 81, 86, 92, 96, 98, 104, 106, 108, 112, 122, 128, 138, 144, 148, 152, 161, 162, 169, 172, 178, 184, 192, 196, 202, 206, 207, 208, 212, 214, 216, 224, 243
COMMENTS
First differs from A331936 in having 69, the Matula-Goebel number of the tree ((o)((o)(o))). A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless the child is an endpoint/leaf.
Locally disjoint means no child of any vertex has branches overlapping the branches of any other (inequivalent) child of the same vertex.
The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of the branches of its root, which gives a bijective correspondence between positive integers and unlabeled rooted trees.
Consists of one, two, and all nonprime numbers whose distinct prime indices are pairwise coprime and already belong to the sequence, where a singleton is always considered to be pairwise coprime. A prime index of n is a number m such that prime(m) divides n.
EXAMPLE
The sequence of all semi-lone-child-avoiding locally disjoint rooted trees together with their Matula-Goebel numbers begins:
1: o
2: (o)
4: (oo)
6: (o(o))
8: (ooo)
9: ((o)(o))
12: (oo(o))
14: (o(oo))
16: (oooo)
18: (o(o)(o))
24: (ooo(o))
26: (o(o(o)))
27: ((o)(o)(o))
28: (oo(oo))
32: (ooooo)
36: (oo(o)(o))
38: (o(ooo))
46: (o((o)(o)))
48: (oooo(o))
49: ((oo)(oo))
MATHEMATICA
msQ[n_]:=n==1||n==2||!PrimeQ[n]&&(PrimePowerQ[n]||CoprimeQ@@PrimePi/@First/@FactorInteger[n])&&And@@msQ/@PrimePi/@First/@FactorInteger[n];
Select[Range[100], msQ]
CROSSREFS
Not requiring lone-child-avoidance gives A316495. The semi-identity tree case is A331681. The non-semi version (i.e., not containing 2) is A331871. These trees counted by vertices are A331872. These trees counted by leaves are A331874. Not requiring local disjointness gives A331935. Cf. A007097, A050381, A061775, A196050, A291636, A302696, A316473, A316696, A316697, A331680, A331682, A331683, A331687, A331934.
Number of lone-child-avoiding rooted semi-identity trees with n vertices.
+10
14
1, 0, 1, 1, 2, 3, 5, 9, 16, 30, 55, 105, 200, 388, 754, 1483, 2923, 5807, 11575, 23190, 46608, 94043, 190287, 386214, 785831, 1602952, 3276845, 6712905, 13778079, 28330583, 58350582, 120370731, 248676129, 514459237, 1065696295, 2210302177, 4589599429, 9540623926
COMMENTS
Lone-child-avoiding means there are no unary branchings.
In a semi-identity tree, the non-leaf branches of any given vertex are distinct.
EXAMPLE
The a(1) = 1 through a(9) = 16 trees (empty column shown as dot):
o . (oo) (ooo) (oooo) (ooooo) (oooooo) (ooooooo) (oooooooo)
(o(oo)) (o(ooo)) (o(oooo)) (o(ooooo)) (o(oooooo))
(oo(oo)) (oo(ooo)) (oo(oooo)) (oo(ooooo))
(ooo(oo)) (ooo(ooo)) (ooo(oooo))
(o(o(oo))) (oooo(oo)) (oooo(ooo))
((oo)(ooo)) (ooooo(oo))
(o(o(ooo))) ((oo)(oooo))
(o(oo(oo))) (o(o(oooo)))
(oo(o(oo))) (o(oo)(ooo))
(o(oo(ooo)))
(o(ooo(oo)))
(oo(o(ooo)))
(oo(oo(oo)))
(ooo(o(oo)))
((oo)(o(oo)))
(o(o(o(oo))))
MATHEMATICA
ssb[n_]:=If[n==1, {{}}, Join@@Function[c, Select[Union[Sort/@Tuples[ssb/@c]], UnsameQ@@DeleteCases[#, {}]&]]/@Rest[IntegerPartitions[n-1]]];
Table[Length[ssb[n]], {n, 10}]
PROG
(PARI) WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
seq(n)={my(v=[0, 0]); for(n=2, n-1, v=concat(v, 1 + vecsum(WeighT(v)) - v[n])); v[1]=1; v} \\ Andrew Howroyd, Feb 09 2020
CROSSREFS
Lone-child-avoiding rooted trees are A001678. The locally disjoint case is A212804. Not requiring lone-child-avoidance gives A306200. Matula-Goebel numbers of these trees are A331965. The semi-lone-child-avoiding version is A331993. Cf. A000081, A004111, A291636, A300660, A306202, A316694, A331683, A331686, A331783, A331875, A331964, A331994.
Lexicographically earliest sequence of positive integers that have at most one distinct prime index already in the sequence.
+10
13
1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 16, 17, 19, 23, 25, 26, 27, 29, 31, 32, 37, 39, 41, 43, 47, 49, 52, 53, 58, 59, 61, 64, 65, 67, 71, 73, 74, 79, 81, 83, 86, 87, 89, 91, 94, 97, 101, 103, 104, 107, 109, 111, 113, 116, 117, 121, 122, 125, 127, 128, 129, 131, 137
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. Conjecture: a(n)/ A331784(n) -> 1 as n -> infinity.
EXAMPLE
The sequence of terms together with their prime indices begins:
1: {} 37: {12} 86: {1,14}
2: {1} 39: {2,6} 87: {2,10}
3: {2} 41: {13} 89: {24}
4: {1,1} 43: {14} 91: {4,6}
5: {3} 47: {15} 94: {1,15}
7: {4} 49: {4,4} 97: {25}
8: {1,1,1} 52: {1,1,6} 101: {26}
9: {2,2} 53: {16} 103: {27}
11: {5} 58: {1,10} 104: {1,1,1,6}
13: {6} 59: {17} 107: {28}
16: {1,1,1,1} 61: {18} 109: {29}
17: {7} 64: {1,1,1,1,1,1} 111: {2,12}
19: {8} 65: {3,6} 113: {30}
23: {9} 67: {19} 116: {1,1,10}
25: {3,3} 71: {20} 117: {2,2,6}
26: {1,6} 73: {21} 121: {5,5}
27: {2,2,2} 74: {1,12} 122: {1,18}
29: {10} 79: {22} 125: {3,3,3}
31: {11} 81: {2,2,2,2} 127: {31}
32: {1,1,1,1,1} 83: {23} 128: {1,1,1,1,1,1,1}
For example, the prime indices of 117 are {2,2,6}, of which only 2 is already in the sequence, so 117 is in the sequence.
MATHEMATICA
aQ[n_]:=Length[Select[PrimePi/@First/@If[n==1, {}, FactorInteger[n]], aQ]]<=1;
Select[Range[100], aQ]
CROSSREFS
Numbers S without all prime indices in S are A324694. Numbers S without any prime indices in S are A324695. Numbers S with at most one prime index in S are A331784. Numbers S with exactly one prime index in S are A331785. Numbers S with exactly one distinct prime index in S are A331913.
Number of lone-child-avoiding locally disjoint unlabeled rooted trees with n vertices.
+10
12
1, 0, 1, 1, 2, 3, 6, 9, 16, 26, 45, 72, 124, 201, 341, 561, 947, 1571, 2651, 4434, 7496, 12631, 21423, 36332, 61910, 105641, 180924, 310548, 534713, 923047
COMMENTS
A rooted tree is locally disjoint if no child of any vertex has branches overlapping the branches of any other unequal child of the same vertex. Lone-child-avoiding means there are no unary branchings.
EXAMPLE
The a(1) = 1 through a(9) = 16 trees (empty column indicated by dot):
o . (oo) (ooo) (oooo) (ooooo) (oooooo) (ooooooo) (oooooooo)
(o(oo)) (o(ooo)) (o(oooo)) (o(ooooo)) (o(oooooo))
(oo(oo)) (oo(ooo)) (oo(oooo)) (oo(ooooo))
(ooo(oo)) (ooo(ooo)) (ooo(oooo))
((oo)(oo)) (oooo(oo)) (oooo(ooo))
(o(o(oo))) (o(o(ooo))) (ooooo(oo))
(o(oo)(oo)) ((ooo)(ooo))
(o(oo(oo))) (o(o(oooo)))
(oo(o(oo))) (o(oo(ooo)))
(o(ooo(oo)))
(oo(o(ooo)))
(oo(oo)(oo))
(oo(oo(oo)))
(ooo(o(oo)))
(o((oo)(oo)))
(o(o(o(oo))))
MATHEMATICA
disjointQ[u_]:=Apply[And, Outer[#1==#2||Intersection[#1, #2]=={}&, u, u, 1], {0, 1}];
strut[n_]:=If[n==1, {{}}, Select[Join@@Function[c, Union[Sort/@Tuples[strut/@c]]]/@Rest[IntegerPartitions[n-1]], disjointQ]];
Table[Length[strut[n]], {n, 10}]
CROSSREFS
The Matula-Goebel numbers of these trees are A331871. The non-locally disjoint version is A001678. These trees counted by number of leaves are A316697. The semi-lone-child-avoiding version is A331872. Cf. A000081, A000669, A005804, A060356, A141268, A300660, A316471, A316473, A316694, A316495, A319312, A330465, A331679, A331681, A331683.
One, two, and all numbers of the form 2^k * prime(j) where k > 0 and j already belongs to the sequence.
+10
12
1, 2, 4, 6, 8, 12, 14, 16, 24, 26, 28, 32, 38, 48, 52, 56, 64, 74, 76, 86, 96, 104, 106, 112, 128, 148, 152, 172, 178, 192, 202, 208, 212, 214, 224, 256, 262, 296, 304, 326, 344, 356, 384, 404, 416, 424, 428, 446, 448, 478, 512, 524, 526, 592, 608, 622, 652
COMMENTS
Also Matula-Goebel numbers of semi-lone-child-avoiding locally disjoint rooted semi-identity trees. A rooted tree is semi-lone-child-avoiding if there are no vertices with exactly one child unless the child is an endpoint/leaf. Locally disjoint means no branch of any vertex overlaps a different (unequal) branch of the same vertex. In a semi-identity tree, all non-leaf branches of any given vertex are distinct. Note that these conditions together imply that there is at most one non-leaf branch under any given vertex.
Also Matula-Goebel numbers of semi-lone-child-avoiding rooted trees with at most one non-leaf branch under any given vertex.
The Matula-Goebel number of a rooted tree is the product of primes indexed by the Matula-Goebel numbers of its branches (of the root), which gives a bijective correspondence between positive integers and unlabeled rooted trees.
EXAMPLE
The sequence of all semi-lone-child-avoiding rooted trees with at most one non-leaf branch under any given vertex, together with their Matula-Goebel numbers, begins:
1: o
2: (o)
4: (oo)
6: (o(o))
8: (ooo)
12: (oo(o))
14: (o(oo))
16: (oooo)
24: (ooo(o))
26: (o(o(o)))
28: (oo(oo))
32: (ooooo)
38: (o(ooo))
48: (oooo(o))
52: (oo(o(o)))
56: (ooo(oo))
64: (oooooo)
74: (o(oo(o)))
76: (oo(ooo))
86: (o(o(oo)))
MAPLE
N:= 1000: # for terms <= N
S:= {1, 2}:
with(queue):
Q:= new(1, 2):
while not empty(Q) do
r:= dequeue(Q);
p:= ithprime(r);
newS:= {seq(2^i*p, i=1..ilog2(N/p))} minus S;
S:= S union newS;
for s in newS do enqueue(Q, s) od:
od:
MATHEMATICA
uryQ[n_]:=n==1||MatchQ[FactorInteger[n], ({{2, _}, {p_, 1}}/; uryQ[PrimePi[p]])|{{2, _}}];
Select[Range[100], uryQ]
CROSSREFS
The enumeration of these trees by nodes is A324969 (essentially A000045). The enumeration of these trees by leaves appears to be A090129(n + 1). The (non-semi) lone-child-avoiding version is A331683. Matula-Goebel numbers of rooted semi-identity trees are A306202. Lone-child-avoiding locally disjoint rooted trees by leaves are A316697. The set S of numbers with at most one prime index in S is A331784. Matula-Goebel numbers of locally disjoint rooted trees are A316495. Cf. A000081, A000669, A001678, A007097, A061775, A196050, A291636, A316470, A316473, A331679, A331680, A331682, A331687, A331783, A331785, A331873.
Lexicographically earliest sequence of positive integers that have at most one prime index already in the sequence, counting multiplicity.
+10
11
1, 2, 3, 5, 7, 11, 13, 14, 17, 19, 21, 23, 26, 29, 31, 35, 37, 38, 39, 41, 43, 46, 47, 49, 53, 57, 58, 59, 61, 65, 67, 69, 71, 73, 74, 77, 79, 83, 87, 89, 91, 94, 95, 97, 98, 101, 103, 106, 107, 109, 111, 113, 115, 119, 122, 127, 131, 133, 137, 139, 141, 142
COMMENTS
A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. Conjecture: A331912(n)/a(n) -> 1 as n -> infinity.
EXAMPLE
The sequence of terms together with their prime indices begins:
1: {} 43: {14} 91: {4,6} 141: {2,15}
2: {1} 46: {1,9} 94: {1,15} 142: {1,20}
3: {2} 47: {15} 95: {3,8} 143: {5,6}
5: {3} 49: {4,4} 97: {25} 145: {3,10}
7: {4} 53: {16} 98: {1,4,4} 147: {2,4,4}
11: {5} 57: {2,8} 101: {26} 149: {35}
13: {6} 58: {1,10} 103: {27} 151: {36}
14: {1,4} 59: {17} 106: {1,16} 157: {37}
17: {7} 61: {18} 107: {28} 158: {1,22}
19: {8} 65: {3,6} 109: {29} 159: {2,16}
21: {2,4} 67: {19} 111: {2,12} 161: {4,9}
23: {9} 69: {2,9} 113: {30} 163: {38}
26: {1,6} 71: {20} 115: {3,9} 167: {39}
29: {10} 73: {21} 119: {4,7} 169: {6,6}
31: {11} 74: {1,12} 122: {1,18} 173: {40}
35: {3,4} 77: {4,5} 127: {31} 178: {1,24}
37: {12} 79: {22} 131: {32} 179: {41}
38: {1,8} 83: {23} 133: {4,8} 181: {42}
39: {2,6} 87: {2,10} 137: {33} 182: {1,4,6}
41: {13} 89: {24} 139: {34} 183: {2,18}
For example, the prime indices of 95 are {3,8}, of which only 3 is in the sequence, so 95 is in the sequence.
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
aQ[n_]:=Length[Cases[primeMS[n], _?aQ]]<=1;
Select[Range[100], aQ]
CROSSREFS
Contains all prime numbers A000040. Numbers S without all prime indices in S are A324694. Numbers S without any prime indices in S are A324695. Numbers S with exactly one prime index in S are A331785. Numbers S with at most one distinct prime index in S are A331912. Numbers S with exactly one distinct prime index in S are A331913.
Number of enriched identity p-trees of weight n.
+10
10
1, 1, 2, 3, 6, 14, 32, 79, 198, 522, 1368, 3716, 9992, 27612, 75692, 212045, 589478, 1668630, 4690792, 13387332, 37980664, 109098556, 311717768, 900846484, 2589449032, 7515759012, 21720369476, 63305262126, 183726039404, 537364221200, 1565570459800, 4592892152163
COMMENTS
An enriched identity p-tree of weight n is either the number n itself or a finite sequence of distinct enriched identity p-trees whose weights are weakly decreasing and sum to n.
EXAMPLE
The a(1) = 1 through a(6) = 14 enriched p-trees:
1 2 3 4 5 6
(21) (31) (32) (42)
((21)1) (41) (51)
((21)2) (321)
((31)1) ((21)3)
(((21)1)1) ((31)2)
((32)1)
(3(21))
((41)1)
((21)21)
(((21)1)2)
(((21)2)1)
(((31)1)1)
((((21)1)1)1)
MATHEMATICA
eptrid[n_]:=Prepend[Select[Join@@Table[Tuples[eptrid/@p], {p, Rest[IntegerPartitions[n]]}], UnsameQ@@#&], n];
Table[Length[eptrid[n]], {n, 10}]
PROG
(PARI) seq(n)={my(v=vector(n)); for(n=1, n, v[n] = 1 + polcoef(prod(k=1, n-1, sum(j=0, n\k, j!*binomial(v[k], j)*x^(k*j)) + O(x*x^n)), n)); v} \\ Andrew Howroyd, Feb 09 2020
CROSSREFS
The locally disjoint case is A331684.
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