John P. McSorley has authored 36 sequences. Here are the ten most recent ones:
A382665
Number of distinct degree sequences among all connected simple graphs with n vertices whose degrees are consecutive integers.
Original entry on oeis.org
1, 1, 1, 2, 5, 14, 35, 88, 212, 492, 1122
Offset: 0
For n = 5 there are 21 non-isomorphic connected graphs G on 5 vertices, and 16 of these have a consecutive degree sequence. However consecutive degree sequences 12223, and 22233 each correspond to 2 non-isomorphic connected graphs. Thus there are 14 distinct graphic sequences of length 5 that are consecutive and represent a connected graph, and so a(5)=14.
- R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford University Press (1999).
A381765
Number of connected simple graphs on n unlabeled vertices whose degree sequence is consecutive.
Original entry on oeis.org
1, 1, 1, 2, 5, 16, 75, 544, 6920, 159228, 6961507, 577826609, 90529308665
Offset: 0
For n = 4 there are 6 non-isomorphic connected graphs G on 4 vertices. An example with consecutive degree sequence is P_4, the path on 4 vertices, with degree sequence 1122; and an example with non-consecutive degree sequence is the star K_{1,3} with degree sequence 1113. All other connected G have consecutive degree sequence. Thus a(4) = 5.
- R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford University Press (1999).
A382021
Number of distinct degree sequences among all simple graphs with n vertices whose degrees are consecutive integers.
Original entry on oeis.org
1, 1, 2, 4, 9, 21, 50, 118, 272, 614, 1368, 3014
Offset: 0
For n = 5 there are 34 non-isomorphic graphs G on 5 vertices, and 24 of these have a consecutive degree sequence. However consecutive degree sequences 11222, 12223, and 22233 each correspond to 2 non-isomorphic graphs. Thus there are 21 distinct consecutive graphic sequences of length 5, and so a(5)=21.
- R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford University Press (1999).
A381586
Number of simple graphs on n unlabeled vertices whose degree sequence is consecutive.
Original entry on oeis.org
1, 1, 2, 4, 9, 24, 98, 622, 7293, 162052, 6997100, 578605618, 90558592724, 26673271109299, 14758661765740616
Offset: 0
For n = 4 there are 11 non-isomorphic graphs G on 4 vertices. An example with consecutive degree sequence is 4K_1, with degree sequence 0000; and an example with non-consecutive degree sequence is K_1 U K_3 with degree sequence 0222. The only other G with non-consecutive degree sequence is K_{1,3} with degree sequence 1113. Thus a(4) = 9.
- R. C. Read and R. J. Wilson, An Atlas of Graphs, Oxford University Press (1999).
A335649
a(n) is the frequency of multi-pairs in a sequence of multi-set designs with 2 blocks.
Original entry on oeis.org
0, 10, 200, 3040, 43320, 607050, 8468880, 118007680, 1643826800, 22896269770, 318906570840, 4441805503200, 61866406977960, 861688028423050, 12001766499380000, 167163044860403200, 2328280868627854560, 32428769142358413450, 451674487223023755240, 6291014052348080593120
Offset: 1
For V={x,y} the design for n=2 are the blocks {xxxxxy,xyyyyy}. Pair frequencies of the multi-pairs xx, yy, and xy in these 2 blocks are all a(2)=10.
A092184(3)=6, and the above example has blocks of size 6.
- A. Assaf, A. Hartman, E. Mendelsohn, Multi-set Designs-Designs having blocks with repeated elements, Congressus Numerantium, 48 (1985), 7-24.
A092184(n+1) is the block size of the n-th design in the sequence.
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LinearRecurrence[{19, -76, 76, -19, 1}, {0, 10, 200, 3040, 43320, 607050}, 20] (* Amiram Eldar, Jun 16 2020 *)
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concat(0, Vec(10*x^2*(1 + x) / ((1 - x)*(1 - 14*x + x^2)*(1 - 4*x + x^2)) + O(x^20))) \\ Colin Barker, Jun 16 2020
A290667
Number of asymmetric equicolorable (unrooted) trees with 2*n vertices.
Original entry on oeis.org
0, 0, 0, 1, 4, 19, 84, 378, 1727, 8126, 39055, 191902, 960681
Offset: 1
a(3) = 0 because there are six trees with 6 vertices, but only three of these have s = t = n = 3, and none of these three is asymmetric. The fourth term a(4) = 1 because there are nine trees with 8 vertices with s = t = n = 4 but only 1 is asymmetric, namely tree T46. See "Atlas of Graphs", page 65.
- R. C. Read and R. J. Wilson, Atlas of Graphs, Oxford Science Publications, Clarendon Press, OUP, 2004.
- F. Hüffner, tinygraph, software for generating integer sequences based on graph properties.
- Austin Mohr, Unlabeled Trees.
Cf.
A119856 (equicolorable trees with 2n vertices),
A000220 (asymmetric trees with n vertices).
a(10)-a(13) added using tinygraph by
Falk Hüffner, Jul 25 2019
A278427
Triangle read by rows: CU(n,k) is the number of unlabeled subgraphs with k edges of the n-cycle C_n.
Original entry on oeis.org
1, 1, 1, 2, 1, 1, 3, 2, 1, 1, 4, 3, 3, 1, 1, 5, 4, 5, 3, 1, 1, 6, 5, 7, 6, 4, 1, 1, 7, 6, 9, 9, 8, 4, 1, 1, 8, 7, 11, 12, 13, 9, 5, 1, 1, 9, 8, 13, 15, 18, 15, 12, 5, 1, 1, 10, 9, 15, 18, 23, 22, 21, 13, 6, 1, 1, 11, 10, 17, 21, 28, 29, 31, 24, 16, 6, 1, 1
Offset: 0
For row n = 3 of the triangle below: there are 3 unlabeled subgraphs of the triangle C_3 with 0 edges, 2 with 1 edge, 1 with 2 edges, and 1 with 3 edges (C_3 itself).
Triangle begins:
1;
1, 1;
2, 1, 1;
3, 2, 1, 1;
4, 3, 3, 1, 1;
5, 4, 5, 3, 1, 1;
6, 5, 7, 6, 4, 1, 1;
7, 6, 9, 9, 8, 4, 1, 1;
8, 7, 11, 12, 13, 9, 5, 1, 1;
9, 8, 13, 15, 18, 15, 12, 5, 1, 1;
10, 9, 15, 18, 23, 22, 21, 13, 6, 1, 1;
...
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\\ here P is A008284 as vector of vectors.
P(n)={[Vecrev(p/y) | p<-Vec(-1 + 1/prod(k=1, n, 1 - y*x^k + O(x*x^n)))]}
T(n)={my(p=P(n-1)); matrix(n, n, n, k, if(k>=n, k==n, sum(i=k, n-1, p[i][i-k+1])))}
{ my(A=T(12)); for(n=1, #A, print(A[n,1..n])) } \\ Andrew Howroyd, Sep 26 2019
Offset corrected and terms a(66) and beyond from
Andrew Howroyd, Sep 26 2019
A277919
Triangle read by rows: CL(n,k) is the number of labeled subgraphs with k edges of the n-cycle C_n.
Original entry on oeis.org
1, 1, 1, 3, 2, 1, 7, 6, 3, 1, 15, 16, 10, 4, 1, 31, 40, 30, 15, 5, 1, 63, 96, 84, 50, 21, 6, 1, 127, 224, 224, 154, 77, 28, 7, 1, 255, 512, 576, 448, 258, 112, 36, 8, 1, 511, 1152, 1440, 1248, 810, 405, 156, 45, 9, 1, 1023, 2560, 3520, 3360, 2420, 1362, 605, 210, 55, 10, 1
Offset: 0
For row 3 of the triangle below: there are 7 labeled subgraphs of the triangle C_3 with 0 edges, 6 with 1 edge, 3 with 2 edges, and 1 with 3 edges (C_3 itself).
Triangle begins:
1;
1, 1;
3, 2, 1;
7, 6, 3, 1;
15, 16, 10, 4, 1;
31, 40, 30, 15, 5, 1;
63, 96, 84, 50, 21, 6, 1;
127, 224, 224, 154, 77, 28, 7, 1;
255, 512, 576, 448, 258, 112, 36, 8, 1;
511, 1152, 1440, 1248, 810, 405, 156, 45, 9, 1;
1023, 2560, 3520, 3360, 2420, 1362, 605, 210, 55, 10, 1;
...
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T(n)={[Vecrev(p) | p<-Vec((1 - 2*x + 2*x^2)/((1-x)*(1 - y*x - 2*x + y*x^2)) + O(x*x^n))]}
{ my(A=T(12)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Sep 27 2019
A266479
Number of n-vertex simple graphs G_n for which n does not divide the number of labeled copies of G_n.
Original entry on oeis.org
0, 2, 2, 6, 3, 20, 4
Offset: 1
If n=3 then both G_3 = K_3 and its complement have a(G_3) = 6, so l(G_3) = 3!/6 = 1, and so 3 does not divide l(G_3); no other graphs G_3 satisfy this, so a(3)=2.
- John P. McSorley, Smallest labelled class (and largest automorphism group) of a tree T_{s,t} and good labellings of a graph, preprint, (2016).
- R. C. Read, R. J. Wilson, An Atlas of Graphs, Oxford Science Publications, Oxford University Press, (1998).
- James Turner, Point-symmetric graphs with a prime number of points, Journal of Combinatorial Theory, vol. 3 (1967), 136-145.
A266478
Number of n-vertex simple graphs G_n for which n divides the number of labeled copies of G_n.
Original entry on oeis.org
1, 0, 2, 5, 31, 136, 1040
Offset: 1
If n=3 then both G_3 = K_1 union K_2 and its complement have a(G_3)=2, so l(G_3) = 3!/2 = 3, and so 3 divides l(G_3); no other graphs G_3 satisfy this, so a(3) = 2.
- John P. McSorley, Smallest labelled class (and largest automorphism group) of a tree T_{s,t} and good labellings of a graph, preprint, (2016).
- R. C. Read, R. J. Wilson, An Atlas of Graphs, Oxford Science Publications, Oxford University Press, (1998).
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