Murray Elder has authored 16 sequences. Here are the ten most recent ones:
A337179
Number of geodetic graphs with n unlabeled vertices.
Original entry on oeis.org
1, 1, 2, 4, 10, 23, 66, 185, 586, 1880, 6360, 21975, 78230, 283087, 1043329, 3895505, 14726263, 56234210, 216719056, 841857211, 3293753840, 12969219563
Offset: 1
For n=4 there are a(4)=4 geodetic graphs: a triangle with another edge attached to one vertex, an edge path of length 3, a tripod of 3 edges joined at a common vertex, and a complete graph on 4 vertices.
o
o o /|\
/ \ | o-|-o
o-o---o, o-o-o-o, o-o-o, \|/
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- John Cu, AMSI Vacation Research Scholar project, 2021
- Brendan McKay and Adolfo Piperno, nauty and Traces. [nauty and Traces are programs for computing automorphism groups of graphs and digraphs.]
- K. R. Parthasarathy and N. Srinivasan, Some general constructions of geodetic blocks, Journal of Combinatorial Theory, Series B (33) Issue 2, October 1982, pp. 121-136.
- Florian Stober and Armin Weiß, Geodetic Graphs: Experiments and New Constructions, arXiv:2308.08970 [math.CO], 2023.
- Eric Weisstein's World of Mathematics, Geodetic Graph
- Wikipedia, Geodetic graph
a(12)-a(22) from Florian Stober and Armin Weiß added by
Murray Elder, Nov 14 2023
A337178
Number of biconnected geodetic graphs with n unlabeled vertices.
Original entry on oeis.org
0, 1, 1, 1, 2, 1, 3, 1, 3, 4, 3, 1, 9, 2, 4, 8, 6, 5, 13, 3, 13, 19, 11, 3, 32
Offset: 1
For n=5 there are exactly a(5)=2 biconnected geodetic graphs: a 5-cycle and the complete graph on 5 vertices.
- John Cu, Length of Embedded Circuits in Geodetic Graphs, AMSI Vacation Research Scholar project, 2021.
- Brendan McKay and Adolfo Piperno, nauty and Traces. [nauty and Traces are programs for computing automorphism groups of graphs and digraphs.]
- K. R. Parthasarathy and N. Srinivasan, Some general constructions of geodetic blocks, Journal of Combinatorial Theory, Series B (33) Issue 2, October 1982, Pages 121-136.
- Florian Stober and Armin Weiß, Geodetic Graphs: Experiments and New Constructions, arXiv:2308.08970 [math.CO], 2023.
- Wikipedia, Geodetic graph
a(12)-a(25) from Florian Stober and Armin Weiß added by
Murray Elder, Nov 14 2023
A246877
Cogrowth sequence for Richard Thompson's group F with the standard generating set x_0, x_1.
Original entry on oeis.org
20, 64, 336, 1160, 5896, 24652, 117628, 531136, 2559552, 12142320, 59416808, 290915560, 1449601452, 7269071976, 36877764000, 188484835300, 972003964976, 5049059855636, 26423287218612, 139205945578944
Offset: 5
The length of the shortest relation in the group presentation is 10, there are 20 distinct cyclic permutations of this word and its inverse, and each one is a reduced trivial word of length 2*5, so a(5)=20.
- Murray Elder, Table of n, a(n) for n = 5..23
- M. Elder, A. Rechnitzer, T. Wong, On the cogrowth of Thompson's group F, Groups, Complexity, Cryptology 4(2) (2012), 301-320.
- S. Haagerup, U. Haagerup, M. Ramirez-Solano, A computational approach to the Thompson group F, Arxiv 2014
A245233
Number of permutations generated by passing an ordered sequence of length n through a stack of depth 2 and an infinite stack in series.
Original entry on oeis.org
1, 1, 2, 6, 24, 114, 592, 3216, 17904, 101198, 578208, 3332136, 19343408, 113017332, 664168160, 3923729280, 23291913440, 138872375958, 831321465408, 4994806458040, 30111586314800, 182094123983660, 1104331746746208, 6715050373394528, 40931670125150624, 250065092876686924, 1530948705125205952, 9391164635349135184
Offset: 0
For n=5 all but 6 permutations can be generated: 51234, 51243, 51423, 52134, 52143, 52413.
- Alois P. Heinz, Table of n, a(n) for n = 0..1000
- M. Elder, Permutations generated by a stack of depth 2 and an infinite stack in series, Electron. J. Combin, 13(1) (2006), R68.
- M. Elder, G. Lee, A. Rechnitzer, Permutations generated by a depth 2 and infinite stack in series are algebraic, arXiv:1407.4248 [math.CO], 2014. Electronic Journal of Combinatorics 22(1) (2015), #P2.16.
- Elizabeth Hartung, Hung Phuc Hoang, Torsten Mütze, Aaron Williams, Combinatorial generation via permutation languages. I. Fundamentals, arXiv:1906.06069 [cs.DM], 2019.
-
a:= proc(n) option remember; `if`(n<5, n!,
(8*(n-2)*(10*n^6+21*n^5-455*n^4-1143*n^3+5227*n^2
+10026*n-1926)*a(n-1) -(400*n^7-1120*n^6-20520*n^5
+56848*n^4+317984*n^3-1096896*n^2+180600*n+939024)*a(n-2)
+(320*n^7-3168*n^6-15520*n^5+198432*n^4+74096*n^3
-3892992*n^2+8591088*n-3756096)*a(n-3) +(2560*n^7
-13824*n^6-108624*n^5+666320*n^4+1015472*n^3-10736624*n^2
+16022304*n-2062944)*a(n-4) -32*(4*n-15)*(4*n-17)*
(2*n-9)*(5*n^4+18*n^3-189*n^2-522*n+2248)*a(n-5)) /
((n-1)*(n+3)*(n+1)*(5*n^4-2*n^3-213*n^2-110*n+2568)))
end:
seq(a(n), n=0..40); # Alois P. Heinz, Jul 14 2014
-
q = (1-2*x-Sqrt[1-4*x])/(2*x); gf = ((1+q)*(1+5*q-q^2-q^3-(1-q)*Sqrt[(1-q^2)*(1-4*q-q^2)] ))/(8*q); CoefficientList[Series[gf, {x, 0, 40}], x] (* Jean-François Alcover, Apr 09 2015, after Joerg Arndt *)
-
N=66; x='x+O('x^N);
q=(1-2*x-sqrt(1-4*x))/(2*x);
gf=((1+q)*(1+5*q-q^2-q^3-(1-q)*sqrt((1-q^2)*(1-4*q-q^2))))/(8*q);
Vec(gf)
\\ Joerg Arndt, Jul 17 2014
A229652
Cogrowth function of the group Baumslag-Solitar(10,10).
Original entry on oeis.org
1, 4, 28, 232, 2092, 19864, 195352, 1970896, 20275660, 211823800, 2240795848, 23951289564, 258255473032, 2805534386256, 30675481454184, 337306578693652, 3727580774618060, 41376921517941032, 461122691909043112, 5157400529078643552
Offset: 0
For n=1 there are 4 words of length 2 equal to the identity: aa^{-1}, a^{-1}a, tt^{-1}, t^{-1}t.
The cogrowth sequences for BS(N,N) for N = 1..10 are
A002894,
A229644,
A229645,
A229646,
A229647,
A229648,
A229649,
A229650,
A229651,
A229652.
A229651
Cogrowth function of the group Baumslag-Solitar(9,9).
Original entry on oeis.org
1, 4, 28, 232, 2092, 19864, 195352, 1970896, 20275660, 211823800, 2240795888, 23951292204, 258255572584, 2805537209648, 30675548482880, 337307986673572, 3727607821613388, 41377406950962504, 461130952671387592, 5157535231753964268
Offset: 0
For n=1 there are 4 words of length 2 equal to the identity: aa^{-1}, a^{-1}a, tt^{-1}, t^{-1}t.
The cogrowth sequences for BS(N,N) for N = 1..10 are
A002894,
A229644,
A229645,
A229646,
A229647,
A229648,
A229649,
A229650,
A229651,
A229652.
A229650
Cogrowth function of the group Baumslag-Solitar(8,8).
Original entry on oeis.org
1, 4, 28, 232, 2092, 19864, 195352, 1970896, 20275660, 211823836, 2240798048, 23951367224, 258257552968, 2805581350056, 30676425237024, 337324008602512, 3727882769574860, 41381900166952348, 461201577710442388, 5158610797198820800
Offset: 0
For n=1 there are 4 words of length 2 equal to the identity: aa^{-1}, a^{-1}a, tt^{-1}, t^{-1}t.
The cogrowth sequences for BS(N,N) for N = 1..10 are
A002894,
A229644,
A229645,
A229646,
A229647,
A229648,
A229649,
A229650,
A229651,
A229652.
A229649
Cogrowth function of the group Baumslag-Solitar(7,7).
Original entry on oeis.org
1, 4, 28, 232, 2092, 19864, 195352, 1970896, 20275692, 211825564, 2240852928, 23952708696, 258285519688, 2806105225928, 30685515254240, 337472968923532, 3730218568024236, 41417273400310152, 461722437389957236, 5166105817092273412
Offset: 0
For n=1 there are 4 words of length 2 equal to the identity: aa^{-1}, a^{-1}a, tt^{-1}, t^{-1}t.
The cogrowth sequences for BS(N,N) for N = 1..10 are
A002894,
A229644,
A229645,
A229646,
A229647,
A229648,
A229649,
A229650,
A229651,
A229652.
A229648
Cogrowth function of the group Baumslag-Solitar(6,6).
Original entry on oeis.org
1, 4, 28, 232, 2092, 19864, 195352, 1970924, 20277036, 211864264, 2241723728, 23969620844, 258583473640, 2811005437348, 30762114003572, 338624821158892, 3747021722921964, 41656518905688504, 465062224305678280, 5211973807553021868
Offset: 0
For n=1 there are 4 words of length 2 equal to the identity: aa^{-1}, a^{-1}a, tt^{-1}, t^{-1}t.
The cogrowth sequences for BS(N,N) for N = 1..10 are
A002894,
A229644,
A229645,
A229646,
A229647,
A229648,
A229649,
A229650,
A229651,
A229652.
A229647
Cogrowth function of the group Baumslag-Solitar(5,5).
Original entry on oeis.org
1, 4, 28, 232, 2092, 19864, 195376, 1971932, 20303084, 212400232, 2251379688, 24129199208, 261067326544, 2848016992032, 31295785633532, 346126420439512, 3850363854970476, 43057199315715676, 483795646775017312, 5459770924922887392
Offset: 0
For n=1 there are 4 words of length 2 equal to the identity: aa^{-1}, a^{-1}a, tt^{-1}, t^{-1}t.
The cogrowth sequences for BS(N,N) for N = 1..10 are
A002894,
A229644,
A229645,
A229646,
A229647,
A229648,
A229649,
A229650,
A229651,
A229652.
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