Journal of Integer Sequences, Vol. 3 (2000), Article 00.1.5 (Tables) |
These tables, which will be updated regularly, include all examples currently known to the author of sequences which occur in the Encyclopedia of Integer Sequences and are U-sequences or L-sequences of oligomorphic permutation groups (in the sense of the accompanying paper Sequences realized by oligomorphic permutation groups).
If an entry is blank, the sequence is not (yet) in the table. As noted in the main paper, all blank entries in the U and L columns should be regarded as potentially interesting sequences.
The column labeled "Inverse Euler" is obtained by applying the transformation EULERi to the U-sequence. Its significance is explained in Section 6 of the main paper. (See also the section on transformations in the On-Line Encyclopedia.) An entry P denotes that the algebra A^{G} for the group in question is known to be polynomial (and the sequence counts its generators).
If the entry under "L-sequence" is annotated with U, then this is the U-sequence of a different group, because the associated Fraïssé class has the Strong Amalgamation Property: see Section 5.4 of the main paper. Such groups are usually not listed separately; the n^{th} term of the L-sequence is obtained by multiplying the n^{th} term of the U-sequence by n! (that is, applying LISTTOLISTMULT).
The letters R and L mean "shifted right" and "shifted left" respectively.
The Atlas of Graphs [5] has been very useful in preparing these tables.
This version is dated 10 September 1999.
In the notation used in [3], A, B, C, D are dC, dC^{*}, C, C^{*} respectively: the "d" denotes point stabilizer and the asterisk denotes an extension by a group of order 2 reversing the order. In the language of species [1], the cases A, B, C, D, S are denoted respectively by L (linear orders), Cha (chains), C (cycles), P (polygons) and E (sets).
Group | U-sequence | L-sequence | Inverse Euler |
---|---|---|---|
S | A000012 | A000012 U | A000007 RP |
A | A000012 | A000142U | A000007RP |
B | A000012 | A001710U | A000007RP |
C | A000012 | A000142RU | A000007RP |
D | A000012 | A001710RU | A000007RP |
Group | U-sequence | L-sequence | Inverse Euler |
---|---|---|---|
S Times S | A000027L | A000079U | (1)P |
S Times A | A000027L | A000522U | (1)P |
A Times A | A000027L | A000142LU | (1)P |
S^{3} | A000217 | A000244U(2) | (1)P |
S^{4} | A000292 | A000302U | (1)P |
S^{5} | A000332 | A000351U | (1)P |
S^{6} | A000389 | A000400U | (1)P |
S^{7} | A000579 | A000420U | (1)P |
S^{8} | A000580 | A001018U | (1)P |
S^{9} | A000581 | A001019U | (1)P |
S^{10} | A000582 | A011557U | (1)P |
S^{11} | A001287 | A001020U | (1)P |
S^{12} | A001288 | A001021U | (1)P |
S^{13} | A001022U | (1)P | |
S^{14} | A001023U | (1)P | |
S^{15} | A001024U | (1)P | |
S^{16} | A001025U | (1)P | |
S^{17} | A001026U | (1)P | |
S^{18} | A001027U | (1)P | |
S^{19} | A001029U | (1)P |
Notes:
Group | U-sequence | L-sequence | Inverse Euler |
---|---|---|---|
S Wr S | A000041 | A000110U | A000012P |
A Wr S | A000041 | A000262U | A000012P |
C Wr S | A000041 | A000142U | A000012P |
S Wr A | A000079R | A000670U | |
S Wr S_{2} | A008619 | A000079U | (1)P |
S_{2} Wr S | A008619 | A000085 | (1)P |
S Wr S_{3} | A001399 | A007051U | (1)P |
S_{3} Wr S | A001399 | A001680 | (1)P |
S Wr S_{4} | A001400 | A007581U | (1)P |
S_{4} Wr S | A001400 | A001681 | (1)P |
S Wr S Wr S | A001970 | A000258U | A000041P |
A Wr A | A000079R | A002866U | A001037 |
A Wr A Wr A(2) | A000244R | U | |
S_{2} Wr A | A000045 | A006206 | |
S_{3} Wr A | A000073 | ||
S_{4} Wr A | A000078 | ||
S_{5} Wr A | A001591 | ||
S_{6} Wr A | A001592 | ||
E Wr S(3) | A002620LL | A000898 | (4)P |
E Wr A(3) | A000129 | (5) | |
S<4>(6) | A007713 | A000307U | A001970P |
S<5> | A007714 | A000357U | A007713P |
S<6> | A000405U | A007714P | |
S<7> | A001669U | P | |
C<2> | A008965 | A003713U | |
C<3> | A000268U | ||
C<4> | A000310U | ||
C<5> | A000359U | ||
C<6> | A000406U | ||
C<7> | A001765U |
Notes:
Group | U-sequence | L-sequence | Inverse Euler |
---|---|---|---|
S Times (S Wr S_{2}) | A002620 | A007051 | (1)P |
S Times (S Wr S_{3}) | A000601 | (1)P | |
S Times (S Wr S_{4}) | A002621 | (1)P | |
S Times (S Wr S_{5}) | A002622 | (1)P | |
S Times (S Wr S) | A000070 | (1)P | |
S Times (S_{2} Wr A) | A000071 | ||
S Wr (S Times S_{2}) | A000097 | A008619P | |
S Times S Times (S Wr S_{2}) | A002623 | (1)P | |
(S_{2} Wr A)^{2} | A001629 | ||
(S_{2} Wr A)^{3} | A001628 | ||
(S_{2} Wr A)^{4} | A001872 | ||
(S_{2} Wr A)^{5} | A001873 | ||
(S_{2} Wr A)^{6} | A001874 | ||
(S_{2} Wr A)^{7} | A001875 |
Notes:
The notation R(things) means that the group is the point stabilizer in the group corresponding to "things" (so that the Fraïssé class is "rooted things"). The L-sequence shifts one place left under this operation.
The treelike objects in this table are discussed in [3]. The final column of the table gives the names used in that paper.
Fraïssé class | U-sequence | L-sequence | Inverse Euler | In [3] |
---|---|---|---|---|
Graphs | A000088 | A006125U | A001349P | |
Graphs up to complement | A007869 | A036442U | ||
K_{3}-free graphs | A006785 | U | A024607P | |
Graphs with bipartite block | A049312 | A047863 | P | |
Graphs with loops | A000666 | A006125LU | P | |
Digraphs | A000273 | (1)U | A003085P | |
Digraphs with loops | A000595 | A002416U | P | |
Oriented graphs | A001174 | A047656U | P | |
Topologies | A001930 | A006057U | A001928P | |
Posets | A000112 | A001035U | A000608P | |
Tournaments | A000568 | A006125U | ||
Local orders | A000016 | A000165RU | C_{2} | |
Two-graphs | A002854 | A006125RU | A003049 | |
Oriented two-graphs | A049313 | A006125RU | ||
Total orders with subset | A000079 | A000165U | A001037P | dC_{2} |
Total orders with 2-partition | A000079R | A002866U | A001037(2)P | dC_{2}^{*} |
C-structures with subset | A000031 | U | ||
D-structures with subset | A000029 | U | ||
2 total orders (distinguished) | A000142 | A001044U | ||
2 total orders (not distinguished) | A007868 | U | ||
2 betweennesses (not distinguished) | A000903 | U | ||
Boron trees (leaves) | A000672RR | A001147RRU | T_{3} | |
HI trees (leaves)(3) | A007827 | A000311RU | T | |
R(Boron trees (leaves)) | A001190 | A001147RU | dT_{3} | |
R(HI trees (leaves))(3) | A000669 | A000311U | dT | |
Trees (edges) | A000055L | A007830RRU | ||
Covington structures[4] | A007866 | A001813RU | dT_{3}(2) | |
Binary trees | A000108 | A001813RU | dPT_{3} | |
Binary trees up to reflection | A007595R | A000407RRU | dP^{*}T_{3} | |
Plane boron trees | A001683 | A001813U | PT_{3} | |
Plane boron trees up to reflection | A000207R | A000407RU | P^{*}T_{3} | |
3-hypergraphs | A000665R | U | A003190P | |
Ternary relations | A000662 | U | P | |
Quaternary relations | A001377 | U | P |
Notes:
Group | U-sequence | L-sequence | Inverse Euler | |
---|---|---|---|---|
(Total orders with subset) Times S | A000225 | U | P | |
(Total orders with subset) Times S Times S | A000295 | U | P | |
(Total orders with subset) Times (Total orders with subset) | A001787 | U | P | |
(Total orders with subset) Times (Total orders with 2-partition) | A001792 | U | P | |
(Total orders with subset) Times (Total orders with 2-partition) Times S | A001787 | U | P | |
(2 total orders (distinguished)) Times S | A003422L | U | ||
R(HI trees (leaves)) Wr S | A000084 | U | A000669P | |
(Binary trees) Wr A | A000108R | U | ||
(Binary trees) Wr A Wr A | A001700 | U |
Group | U-sequence | L-sequence | Inverse Euler |
---|---|---|---|
S on 2-sets | A000664 | A014500 | A002905P |
S^{2} (product action) | A049311 | (1)P |
Notes:
The groups C Wr S and A have the same L-sequence, namely A000142 (factorials). This reflects the fact that a permutation can be expressed uniquely as an unordered union of cycles. Of course, C Wr S has the same U-sequence as S Wr S, namely A000041 (partitions).
Received September 2 1999; revised version received January 4 2000. Published in Journal of Integer Sequences January 25 2000.