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1, 2, 6, 20, 66, 212, 666, 2060, 6306, 19172, 58026, 175100, 527346, 1586132, 4766586, 14316140, 42981186, 129009092, 387158346, 1161737180, 3485735826, 10458256052, 31376865306, 94134790220, 282412759266, 847255055012
COMMENTS
Binomial transform of A000225 (if this starts 1,1,3,7....). Let P(A) be the power set of an n-element set A. Then a(n) = the number of pairs of elements {x,y} of P(A) for which either 0) x and y are intersecting and for which either x is a proper subset of y or y is a proper subset of x, or 1) x = y. - Ross La Haye, Jan 10 2008 Let P(A) be the power set of an n-element set A and R be a relation on P(A) such that for all x, y of P(A), xRy if either 0) x is not a subset of y and y is not a subset of x and x and y are disjoint, or 1) x equals y. Then a(n) = |R|. - Ross La Haye, Mar 19 2009
FORMULA
G.f.: (1-4*x+5*x^2)/((1-x)*(1-2*x)*(1-3*x)).
E.g.f.: exp(3*x) - exp(2*x) + exp(x).
a(n) = 2*StirlingS2(n+1,3) + StirlingS2(n+1,2) + 1. - Ross La Haye, Jan 10 2008
EXAMPLE
Also the number of achiral set-systems on n vertices, where a set-system is achiral if it is not changed by any permutation of the covered vertices. For example, the a(0) = 1 through a(3) = 20 achiral set-systems are:
0 0 0 0
{1} {1} {1}
{2} {2}
{12} {3}
{1}{2} {12}
{1}{2}{12} {13}
{23}
{123}
{1}{2}
{1}{3}
{2}{3}
{1}{2}{3}
{1}{2}{12}
{1}{3}{13}
{2}{3}{23}
{12}{13}{23}
{1}{2}{3}{123}
{12}{13}{23}{123}
{1}{2}{3}{12}{13}{23}
{1}{2}{3}{12}{13}{23}{123}
BII-numbers of these set-systems are A330217. Fully chiral set-systems are A330282, with covering case A330229. (End)
MATHEMATICA
LinearRecurrence[{6, -11, 6}, {1, 2, 6}, 30] (* G. C. Greubel, Feb 13 2019 *)
PROG
(Sage) [3^n-2^n+1 for n in range(30)] # G. C. Greubel, Feb 13 2019 (GAP) List([0..30], n -> 3^n-2^n+1); # G. C. Greubel, Feb 13 2019
Number of non-isomorphic achiral multiset partitions of weight n.
+10
15
1, 1, 4, 5, 12, 9, 30, 17, 52, 44
COMMENTS
A multiset partition is a finite multiset of finite nonempty multisets. It is achiral if it is not changed by any permutation of the vertices.
EXAMPLE
Non-isomorphic representatives of the a(1) = 1 through a(5) = 9 multiset partitions:
{1} {11} {111} {1111} {11111}
{12} {123} {1122} {12345}
{1}{1} {1}{11} {1234} {1}{1111}
{1}{2} {1}{1}{1} {1}{111} {11}{111}
{1}{2}{3} {11}{11} {1}{1}{111}
{11}{22} {1}{11}{11}
{12}{12} {1}{1}{1}{11}
{1}{1}{11} {1}{1}{1}{1}{1}
{1}{2}{12} {1}{2}{3}{4}{5}
{1}{1}{1}{1}
{1}{1}{2}{2}
{1}{2}{3}{4}
Non-isomorphic representatives of the a(6) = 30 multiset partitions:
{111111} {1}{11111} {1}{1}{1111} {1}{1}{1}{111} {1}{1}{1}{1}{11}
{111222} {11}{1111} {1}{11}{111} {1}{1}{11}{11} {1}{1}{2}{2}{12}
{112233} {111}{111} {11}{11}{11} {1}{2}{11}{22}
{123456} {111}{222} {11}{12}{22} {1}{2}{12}{12}
{112}{122} {11}{22}{33} {1}{2}{3}{123} {1}{1}{1}{1}{1}{1}
{12}{1122} {1}{2}{1122} {1}{1}{1}{2}{2}{2}
{123}{123} {12}{12}{12} {1}{1}{2}{2}{3}{3}
{12}{13}{23} {1}{2}{3}{4}{5}{6}
CROSSREFS
Achiral set-systems are counted by A083323. BII-numbers of achiral set-systems are A330217. Achiral integer partitions are counted by A330224. Non-isomorphic fully chiral multiset partitions are A330227. MM-numbers of achiral multisets of multisets are A330232. Achiral factorizations are A330234.
Number of fully chiral set-systems covering n vertices.
+10
14
COMMENTS
A set-system is a finite set of finite nonempty sets. It is fully chiral if every permutation of the vertices gives a different representative.
EXAMPLE
The a(3) = 42 set-systems:
{1}{2}{13} {1}{2}{12}{13} {1}{2}{12}{13}{123}
{1}{2}{23} {1}{2}{12}{23} {1}{2}{12}{23}{123}
{1}{3}{12} {1}{3}{12}{13} {1}{3}{12}{13}{123}
{1}{3}{23} {1}{3}{13}{23} {1}{3}{13}{23}{123}
{2}{3}{12} {2}{3}{12}{23} {2}{3}{12}{23}{123}
{2}{3}{13} {2}{3}{13}{23} {2}{3}{13}{23}{123}
{1}{12}{23} {1}{2}{13}{123}
{1}{13}{23} {1}{2}{23}{123}
{2}{12}{13} {1}{3}{12}{123}
{2}{13}{23} {1}{3}{23}{123}
{3}{12}{13} {2}{3}{12}{123}
{3}{12}{23} {2}{3}{13}{123}
{1}{12}{123} {1}{12}{23}{123}
{1}{13}{123} {1}{13}{23}{123}
{2}{12}{123} {2}{12}{13}{123}
{2}{23}{123} {2}{13}{23}{123}
{3}{13}{123} {3}{12}{13}{123}
{3}{23}{123} {3}{12}{23}{123}
MATHEMATICA
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]], i}, {i, Length[p]}])], {p, Permutations[Union@@m]}]];
Table[Length[Select[Subsets[Subsets[Range[n], {1, n}]], Union@@#==Range[n]&&Length[graprms[#]]==n!&]], {n, 0, 3}]
CROSSREFS
The non-covering version is A330282. Costrict (or T_0) covering set-systems are A059201. BII-numbers of fully chiral set-systems are A330226. Non-isomorphic, fully chiral multiset partitions are A330227. Fully chiral partitions are counted by A330228. Fully chiral covering set-systems are A330229. Fully chiral factorizations are A330235. MM-numbers of fully chiral multisets of multisets are A330236.
MM-numbers of achiral multisets of multisets.
+10
14
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 36, 38, 40, 41, 42, 43, 44, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 66, 67, 68, 72, 73, 76, 79, 80
COMMENTS
First differs from A322554 in lacking 141. A multiset of multisets is achiral if it is not changed by any permutation of the vertices.
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. The multiset of multisets with MM-number n is formed by taking the multiset of prime indices of each part of the multiset of prime indices of n. For example, the prime indices of 78 are {1,2,6}, so the multiset of multisets with MM-number 78 is {{},{1},{1,2}}.
EXAMPLE
The sequence of non-achiral multisets of multisets (the complement of this sequence) together with their MM-numbers begins:
35: {{2},{1,1}}
37: {{1,1,2}}
39: {{1},{1,2}}
45: {{1},{1},{2}}
61: {{1,2,2}}
65: {{2},{1,2}}
69: {{1},{2,2}}
70: {{},{2},{1,1}}
71: {{1,1,3}}
74: {{},{1,1,2}}
75: {{1},{2},{2}}
77: {{1,1},{3}}
78: {{},{1},{1,2}}
87: {{1},{1,3}}
89: {{1,1,1,2}}
90: {{},{1},{1},{2}}
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Apply[Rule, Table[{p[[i]], i}, {i, Length[p]}], {1}])], {p, Permutations[Union@@m]}]]
Select[Range[100], Length[graprms[primeMS/@primeMS[#]]]==1&]
CROSSREFS
The fully-chiral version is A330236. Achiral set-systems are counted by A083323. MG-numbers of planted achiral trees are A214577. MM-numbers of costrict (or T_0) multisets of multisets are A322847. BII-numbers of achiral set-systems are A330217. Non-isomorphic achiral multiset partitions are A330223. Achiral integer partitions are counted by A330224. Achiral factorizations are A330234.
BII-numbers of achiral set-systems.
+10
13
0, 1, 2, 3, 4, 7, 8, 9, 10, 11, 16, 25, 32, 42, 52, 63, 64, 75, 116, 127, 128, 129, 130, 131, 136, 137, 138, 139, 256, 385, 512, 642, 772, 903, 1024, 1155, 1796, 1927, 2048, 2184, 2320, 2457, 2592, 2730, 2868, 3007, 4096, 4233, 6416, 6553, 8192, 8330
COMMENTS
A set-system is a finite set of finite nonempty sets. It is achiral if it is not changed by any permutation of the vertices.
A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793. We define the set-system with BII-number n to be obtained by taking the binary indices of each binary index of n. Every set-system has a different BII-number. For example, 18 has reversed binary expansion (0,1,0,0,1), and since the binary indices of 2 and 5 are {2} and {1,3} respectively, the BII-number of {{2},{1,3}} is 18. Elements of a set-system are sometimes called edges.
EXAMPLE
The sequence of all achiral set-systems together with their BII-numbers begins:
1: {{1}}
2: {{2}}
3: {{1},{2}}
4: {{1,2}}
7: {{1},{2},{1,2}}
8: {{3}}
9: {{1},{3}}
10: {{2},{3}}
11: {{1},{2},{3}}
16: {{1,3}}
25: {{1},{3},{1,3}}
32: {{2,3}}
42: {{2},{3},{2,3}}
52: {{1,2},{1,3},{2,3}}
63: {{1},{2},{3},{1,2},{1,3},{2,3}}
64: {{1,2,3}}
75: {{1},{2},{3},{1,2,3}}
MATHEMATICA
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]], i}, {i, Length[p]}])], {p, Permutations[Union@@m]}]];
Select[Range[0, 1000], Length[graprms[bpe/@bpe[#]]]==1&]
CROSSREFS
These are numbers n such that A330231(n) = 1. Achiral set-systems are counted by A083323. MG-numbers of planted achiral trees are A214577. Non-isomorphic achiral multiset partitions are A330223. Achiral integer partitions are counted by A330224. BII-numbers of fully chiral set-systems are A330226. MM-numbers of achiral multisets of multisets are A330232. Achiral factorizations are A330234. Cf. A000120, A003238, A016031, A048793, A070939, A326031, A326702, A327080, A327081, A330218, A330229, A330233.
Number of fully chiral factorizations of n.
+10
10
1, 1, 1, 2, 1, 0, 1, 3, 2, 0, 1, 4, 1, 0, 0, 5, 1, 4, 1, 4, 0, 0, 1, 7, 2, 0, 3, 4, 1, 0, 1, 7, 0, 0, 0, 4, 1, 0, 0, 7, 1, 0, 1, 4, 4, 0, 1, 12, 2, 4, 0, 4, 1, 7, 0, 7, 0, 0, 1, 4, 1, 0, 4, 11, 0, 0, 1, 4, 0, 0, 1, 16, 1, 0, 4, 4, 0, 0, 1, 12, 5, 0, 1, 4, 0, 0
COMMENTS
A multiset of multisets is fully chiral every permutation of the vertices gives a different representative. A factorization is fully chiral if taking the multiset of prime indices of each factor gives a fully chiral multiset of multisets.
EXAMPLE
The a(n) factorizations for n = 1, 4, 8, 12, 16, 24, 48:
() (4) (8) (12) (16) (24) (48)
(2*2) (2*4) (2*6) (2*8) (3*8) (6*8)
(2*2*2) (3*4) (4*4) (4*6) (2*24)
(2*2*3) (2*2*4) (2*12) (3*16)
(2*2*2*2) (2*2*6) (4*12)
(2*3*4) (2*3*8)
(2*2*2*3) (2*4*6)
(3*4*4)
(2*2*12)
(2*2*2*6)
(2*2*3*4)
(2*2*2*2*3)
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
facs[n_]:=If[n<=1, {{}}, Join@@Table[Map[Prepend[#, d]&, Select[facs[n/d], Min@@#>=d&]], {d, Rest[Divisors[n]]}]];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]], i}, {i, Length[p]}])], {p, Permutations[Union@@m]}]];
Table[Length[Select[facs[n], Length[graprms[primeMS/@#]]==Length[Union@@primeMS/@#]!&]], {n, 100}]
CROSSREFS
The costrict (or T_0) version is A316978. BII-numbers of fully chiral set-systems are A330226. Non-isomorphic fully chiral multiset partitions are A330227. Full chiral partitions are A330228. Fully chiral covering set-systems are A330229. MM-numbers of fully chiral multisets of multisets are A330236.
Number of achiral integer partitions of n.
+10
9
1, 1, 2, 3, 5, 7, 11, 13, 18, 21, 30, 32, 43, 46, 57, 64, 79, 83, 103, 107, 130, 141, 162, 171, 205, 214, 245, 258, 297, 307, 357, 373, 423, 441, 493, 513, 591, 607, 674, 702, 790, 817, 917, 938, 1040, 1078, 1186, 1216, 1362, 1395, 1534, 1580, 1738, 1779, 1956
COMMENTS
A multiset of multisets is achiral if it is not changed by any permutation of the vertices. An integer partition is achiral if taking the multiset of prime indices of each part gives an achiral multiset of multisets.
EXAMPLE
The a(1) = 1 through a(7) = 13 partitions:
(1) (2) (3) (4) (5) (6) (7)
(11) (21) (22) (32) (33) (52)
(111) (31) (41) (42) (61)
(211) (221) (51) (331)
(1111) (311) (222) (421)
(2111) (321) (511)
(11111) (411) (2221)
(2211) (3211)
(3111) (4111)
(21111) (22111)
(111111) (31111)
(211111)
(1111111)
MATHEMATICA
primeMS[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]], i}, {i, Length[p]}])], {p, Permutations[Union@@m]}]];
Table[Length[Select[IntegerPartitions[n], Length[graprms[primeMS/@#]]==1&]], {n, 0, 30}]
CROSSREFS
The fully-chiral version is A330228. The Heinz numbers of these partitions are given by A330232. Achiral set-systems are counted by A083323. BII-numbers of achiral set-systems are A330217. Non-isomorphic achiral multiset partitions are A330223. Achiral factorizations are A330234.
Number of fully chiral set-systems on n vertices.
+10
6
COMMENTS
A set-system is a finite set of finite nonempty sets. It is fully chiral if every permutation of the covered vertices gives a different representative.
EXAMPLE
The a(0) = 1 through a(2) = 5 set-systems:
{} {} {}
{{1}} {{1}}
{{2}}
{{1},{1,2}}
{{2},{1,2}}
MATHEMATICA
graprms[m_]:=Union[Table[Sort[Sort/@(m/.Rule@@@Table[{p[[i]], i}, {i, Length[p]}])], {p, Permutations[Union@@m]}]];
Table[Length[Select[Subsets[Subsets[Range[n], {1, n}]], Length[graprms[#]]==Length[Union@@#]!&]], {n, 0, 3}]
CROSSREFS
Costrict (or T_0) set-systems are A326940. BII-numbers of fully chiral set-systems are A330226. Non-isomorphic fully chiral multiset partitions are A330227. Fully chiral partitions are A330228. Fully chiral factorizations are A330235. MM-numbers of fully chiral multisets of multisets are A330236.
Number of non-isomorphic fully chiral set-systems on n vertices.
+10
5
COMMENTS
A set-system is a finite set of finite nonempty sets. It is fully chiral if every permutation of the covered vertices gives a different representative.
EXAMPLE
Non-isomorphic representatives of the a(0) = 1 through a(3) = 10 set-systems:
0 0 0 0
{1} {1} {1}
{2}{12} {2}{12}
{1}{3}{23}
{2}{13}{23}
{3}{23}{123}
{2}{3}{13}{23}
{1}{3}{23}{123}
{2}{13}{23}{123}
{2}{3}{13}{23}{123}
CROSSREFS
Partial sums of A330295 (the covering case). Unlabeled costrict (or T_0) set-systems are A326946. BII-numbers of fully chiral set-systems are A330226. Non-isomorphic fully chiral multiset partitions are A330227. Fully chiral partitions are A330228. Fully chiral factorizations are A330235. MM-numbers of fully chiral multisets of multisets are A330236.
Number of non-isomorphic fully chiral set-systems covering n vertices.
+10
5
COMMENTS
A set-system is a finite set of finite nonempty sets. It is fully chiral if every permutation of the covered vertices gives a different representative.
EXAMPLE
Non-isomorphic representatives of the a(0) = 1 through a(3) = 7 set-systems:
0 {1} {1}{12} {1}{2}{13}
{1}{12}{23}
{1}{12}{123}
{1}{2}{12}{13}
{1}{2}{13}{123}
{1}{12}{23}{123}
{1}{2}{12}{13}{123}
CROSSREFS
First differences of A330294 (the non-covering case). Unlabeled costrict (or T_0) set-systems are A326946. BII-numbers of fully chiral set-systems are A330226. Non-isomorphic fully chiral multiset partitions are A330227. Fully chiral partitions are A330228. Fully chiral factorizations are A330235. MM-numbers of fully chiral multisets of multisets are A330236. Cf. A000612, A016031, A055621, A083323, A283877, A319637, A330098, A330231, A330232, A330234, A330282.
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