Aziza Jefferson - cp's OEIS frontend

User: Aziza Jefferson

Aziza Jefferson has authored 9 sequences.

A256338 Number of A&U Family matchings on n edges.

1, 3, 14, 81, 526, 3655, 26522, 198322, 1516296, 11794717, 93028387, 742192059, 5978650560, 48558821234, 397218622275, 3269629207524, 27061726430000, 225078993453143, 1880240716499975, 15768890757767329, 132719696885282352, 1120664726059889642, 9490737694928103944, 80593740187789336604, 686097231181385302494, 5854230604182513256777, 50058728487687099021228, 428893610758038945556024, 3681458291424994103104272, 31654643493605098603330050, 272617697673293256259943417, 2351397730980411031399548438, 20310185543805378949877753778, 175663385844074502933143530174, 1521230708939544454165789841800, 13189400713003422051741601456307, 114483609078595784724427186310842, 994773380472692869438699360298740, 8652545469871591210786412806190538

Offset: 1

Aziza Jefferson, Mar 29 2015

nonn

The A&U Family of matchings is the largest family of matchings formed by pinning edges to the right, edge inflation by ladders and vertex insertions.

			a(4)=81 because of the 105 matchings on 4 edges which can be drawn in the plane, 24 do not lie in the A&U Family. Of these 24, only three lie in the R&E family. In canonical sequence form the three missing matchings are given by 12134324, 12324314, and 12343142.

T. Akutsu, Dynamic programming algorithms for RNA secondary structure prediction with pseudoknots, Discrete Appl. Math. 104(1-2), (2000), 45-62.
A. Condon, B. Davy, B. Rastegari, S. Zhao and F. Tarrant, RNA pseudoknotted structures, Theoret. Comput. Sci. 320(1), (2004), 35-50.
Aziza Jefferson, The Substitution Decomposition of Matchings and RNA Secondary Structures, PhD Thesis, University of Florida, 2015.
C. Saule, M. Régnier, J.-M. Steyaert, and A. Denise, Counting RNA pseudoknotted structures, J. Comput. Biol. 18(10), (2011), 1339-1351.
Y. Uemura, A. Hasegawa, S. Kobayashi, and T. Yokomori, Tree adjoining grammars for RNA structure prediction, Theoret. Comput. Sci. 210(2), (1999), 277-303.

Cf. A256330, A256337.

f := RootOf(2*x^7*_Z^(14)-2*x^6*_Z^(13)-3*x^6*_Z^(12)+7*x^5*_Z^(11)+3*x^5*_Z^(10)-16*x^4*_Z^9+2*x^4*_Z^8+18*x^3*_Z^7-7*x^2*_Z^5+(-12*x^3+2*x^2)*_Z^6+(4*x^2-5*x)*_Z^4+10*x*_Z^3+(-5*x+1)*_Z^2-2*_Z+1); convert(series(f, x=0, 40), radical);

G.f. f satisfies 2x^7f^14 - 2x^6f^13 - 3x^6f^12 + 7x^5f^11 + 3x^5f^10 - 16x^4f^9 + 2x^4f^8 + 18x^3f^7 - 7x^2f^5 + (-12x^3 + 2x^2)f^6 + (4x^2 - 5x)f^4 + 10xf^3 + (-5x+1)f^2 - 2f + 1 = 0.

A256337 Number of R&E Family matchings on n edges.

Original entry on oeis.org

1, 3, 14, 84, 592, 4624, 38527, 334520, 2985938, 27183525, 251204717, 2349231434, 22186989871, 211295835831, 2026765351330, 19563183525857, 189878734136185, 1852021863338181, 18143610295356357, 178450501118284066, 1761425423533593329, 17443012946883397306, 173247241040324621443, 1725404763442479917997, 17226646104539624029186, 172389875494663129310211, 1728819399958251503320772, 17371923980126335814068340, 174882805619639894274925366, 1763573883561574064590764255, 17813088298000097109198747848, 180193867968101208748329431620, 1825392351254024651444300421782, 18516189068195200519689971895953, 188058513067187124022632768762920, 1912273561123769149363329826421051, 19466808894886697821847017471202077, 198381665775949932177580813656191187, 2023702201274806730119560173885757279, 20663697938590662344370538856632433182

Offset: 1

Aziza Jefferson, Mar 29 2015

nonn

The R&E Family of matchings is the largest family of matchings formed by repeated edge pinnings, edge inflations by ladders and vertex insertions.

			a(6)=4624 because, of the 4659 matchings on 6 edges which can be drawn in the plane, 65 do not lie in the R&E Family. In canonical sequence form, two of these missing matchings are given by 123143546526 and 123145643625.

A. Condon, B. Davy, B. Rastegari, S. Zhao and F. Tarrant, RNA pseudoknotted structures, Theoret. Comput. Sci. 320(1), (2004), 35-50.
Aziza Jefferson, The Substitution Decomposition of Matchings and RNA Secondary Structures, PhD Thesis, University of Florida, 2015.
E. Rivas and S.R. Eddy, A dynamic programing algorithm for RNA structure prediction including pseudoknots, J. Mol. Biol., 285, (1999), 2053-2068.

Cf. A256330, A256338

f := RootOf(2*x^(30)*_Z^(60) - 2*x^(29)*_Z^(59) - 38*x^(29)*_Z^(58) + 43*x^(28)*_Z^(57) + 342*x^(28)*_Z^(56) - 452*x^(27)*_Z^(55) - 1952*x^(27)*_Z^(54) + 3104*x^(26)*_Z^(53) + 8030*x^(26)*_Z^(52) - 15740*x^(25)*_Z^(51) - 25796*x^(25)*_Z^(50) + 63234*x^(24)*_Z^(49) + 68385*x^(24)*_Z^(48) - 210718*x^(23)*_Z^(47) - 154085*x^(23)*_Z^(46) + 600645*x^(22)*_Z^(45) + 295081*x^(22)*_Z^(44) - 1493939*x^(21)*_Z^(43) - (465768*x^(21)-30*x^(20))*_Z^(42) + 3281450*x^(20)*_Z^(41) + (556761*x^(20)-535*x^(19))*_Z^(40) - 6407159*x^(19)*_Z^(39) - (345063*x^(19)-4503*x^(18))*_Z^(38) + 11148795*x^(18)*_Z^(37) - (486467*x^(18)+23656*x^(17))*_Z^(36) - 17269156*x^(17)*_Z^(35) + (2247622*x^(17)+86500*x^(16))*_Z^(34) + 23701163*x^(16)*_Z^(33) - (5008690*x^(16)+232556*x^(15))*_Z^(32) - (28606954*x^(15)-2*x^(14))*_Z^(31) + (8286505*x^(15)+473426*x^(14))*_Z^(30) + (30112233*x^(14)-33*x^(13))*_Z^(29) - (10981985*x^(14)+739870*x^(13))*_Z^(28) - (27473085*x^(13)-241*x^(12))*_Z^(27) + (11935909*x^(13)+887762*x^(12))*_Z^(26) + (21720351*x^(12)-1046*x^(11))*_Z^(25) - (10798494*x^(12)+802751*x^(11))*_Z^(24) - (15014115*x^(11)-3036*x^(10))*_Z^(23) + (8290091*x^(11)+514052*x^(10))*_Z^(22) + (9236626*x^(10)-6259*x^(9))*_Z^(21) - (5538246*x^(10)+180970*x^(9))*_Z^(20) - (5149642*x^(9)-9471*x^(8))*_Z^(19) + (3290018*x^(9)-42243*x^(8))*_Z^(18) + (2610905*x^(8)-10692*x^(7))*_Z^(17) - (1741070*x^(8)-114094*x^(7))*_Z^(16) - (1180928*x^(7)-9042*x^(6))*_Z^(15) + (800928*x^(7)-91214*x^(6))*_Z^(14) + (460434*x^(6)-5687*x^(5))*_Z^(13) - (307889*x^(6)-46482*x^(5))*_Z^(12) - (148829*x^(5)-2607*x^(4))*_Z^(11) + (94979*x^(5)-16566*x^(4))*_Z^(10) + (38177*x^(4)-838*x^(3))*_Z^(9) - (22576*x^(4)-4140*x^(3))*_Z^(8) - (7344*x^(3)-176*x^(2))*_Z^(7) + (3919*x^(3)-695*x^(2))*_Z^(6) + (981*x^(2)-21*x)*_Z^(5) - (458*x^(2)-70*x)*_Z^(4) - (81*x-1)*_Z^(3) + (32*x-3)*_Z^(2) + 3*_Z - 1, 1);series(f, x=0, 40);

G.f. f satisfies 2*x^(30)*f^(60) - 2*x^(29)*f^(59) - 38*x^(29)*f^(58) + 43*x^(28)*f^(57) + 342*x^(28)*f^(56) - 452*x^(27)*f^(55) - 1952*x^(27)*f^(54) + 3104*x^(26)*f^(53) + 8030*x^(26)*f^(52) - 15740*x^(25)*f^(51) - 25796*x^(25)*f^(50) + 63234*x^(24)*f^(49) + 68385*x^(24)*f^(48) - 210718*x^(23)*f^(47) - 154085*x^(23)*f^(46) + 600645*x^(22)*f^(45) + 295081*x^(22)*f^(44) - 1493939*x^(21)*f^(43) - (465768*x^(21)-30*x^(20))*f^(42) + 3281450*x^(20)*f^(41) + (556761*x^(20)-535*x^(19))*f^(40) - 6407159*x^(19)*f^(39) - (345063*x^(19)-4503*x^(18))*f^(38) + 11148795*x^(18)*f^(37) - (486467*x^(18)+23656*x^(17))*f^(36) - 17269156*x^(17)*f^(35) + (2247622*x^(17)+86500*x^(16))*f^(34) + 23701163*x^(16)*f^(33) - (5008690*x^(16)+232556*x^(15))*f^(32) - (28606954*x^(15)-2*x^(14))*f^(31) + (8286505*x^(15)+473426*x^(14))*f^(30) + (30112233*x^(14)-33*x^(13))*f^(29) - (10981985*x^(14)+739870*x^(13))*f^(28) - (27473085*x^(13)-241*x^(12))*f^(27) + (11935909*x^(13)+887762*x^(12))*f^(26) + (21720351*x^(12)-1046*x^(11))*f^(25) - (10798494*x^(12)+802751*x^(11))*f^(24) - (15014115*x^(11)-3036*x^(10))*f^(23) + (8290091*x^(11)+514052*x^(10))*f^(22) + (9236626*x^(10)-6259*x^(9))*f^(21) - (5538246*x^(10)+180970*x^(9))*f^(20) - (5149642*x^(9)-9471*x^(8))*f^(19) + (3290018*x^(9)-42243*x^(8))*f^(18) + (2610905*x^(8)-10692*x^(7))*f^(17) - (1741070*x^(8)-114094*x^(7))*f^(16) - (1180928*x^(7)-9042*x^(6))*f^(15) + (800928*x^(7)-91214*x^(6))*f^(14) + (460434*x^(6)-5687*x^(5))*f^(13) - (307889*x^(6)-46482*x^(5))*f^(12) - (148829*x^(5)-2607*x^(4))*f^(11) + (94979*x^(5)-16566*x^(4))*f^(10) + (38177*x^(4)-838*x^(3))*f^(9) - (22576*x^(4)-4140*x^(3))*f^(8) - (7344*x^(3)-176*x^(2))*f^(7) + (3919*x^(3)-695*x^(2))*f^(6) + (981*x^(2)-21*x)*f^(5) - (458*x^(2)-70*x)*f^(4) - (81*x-1)*f^(3) + (32*x-3)*f^(2) + 3*f - 1 = 0.

A256336 Number of Largest Chain Ladder Family (LCLF) matchings on n edges.

Original entry on oeis.org

1, 3, 14, 81, 521, 3554, 25172, 183129, 1359863, 10264359, 78521474, 607449380, 4744167924, 37355679904, 296232263792, 2363773540473, 18965408058723, 152910824717297, 1238260516988018, 10066874219853977, 82134185988563049, 672294915226393926, 5519252917557226452

Offset: 1

Aziza Jefferson, Mar 25 2015

nonn

The Largest Chain Ladder Family (LCLF) of matchings is the largest family of matchings formed by repeated edge inflations by ladders and vertex insertions into a chain of any length.

			a(3)=14 because of the 15 matchings on 3 edges, only one does not lie in the Largest Chain Ladder Family. In canonical sequence form, the missing matching is given by 123123.

Aziza Jefferson, The Substitution Decomposition of Matchings and RNA Secondary Structures, PhD Thesis, University of Florida, 2015.

f := RootOf(2*x^3*_Z^6-2*x^2*_Z^5+4*x^2*_Z^4-3*x*_Z^3+2*x*_Z^2+_Z-1,1);
series(f, x=0, 30);

G.f. f satisfies 2x^3f^6-2x^2f^5+4x^2f^4-3xf^3+2xf^2+f-1=0.

A256335 Number of Largest Chain Family matchings on n edges.

Original entry on oeis.org

1, 3, 15, 93, 639, 4670, 35607, 280069, 2255979, 18516875, 154313881, 1302252294, 11106135906, 95571461319, 828803505465, 7235996887013, 63549647848195, 561049960940540, 4976419846070007, 44325237810194705, 396301576614077927, 3555397444230816343, 31996727212476905751, 288776859922595203094, 2613107152879937592054, 23702850369539462227046, 215483061767106353850246, 1963017891713523908516093, 17917224620763719834090179, 163830901587493323034301583, 1500542646711279198177939831, 13765184019931774406496702885

Offset: 1

Aziza Jefferson, Mar 25 2015

nonn

The Largest Chain Family of matchings is the largest family of matchings formed by repeated edge inflations and vertex insertions into any length n chain.

			a(4)=93 because of the 105 matchings on 4 edges, there are 13 matchings which do not lie in the Largest Chain Family. Two such matching in canonical sequence form, are given by 12343142 and 12342413.

Aziza Jefferson, The Substitution Decomposition of Matchings and RNA Secondary Structures, PhD Thesis, University of Florida, 2015.

f := RootOf(_Z^6*x^3+_Z^5*x^2-4*_Z^4*x^2+2*_Z^3*x+_Z^2*x+4*_Z^2-11*_Z+7, 1);
series(f, x=0, 30);

G.f. f satisfies x^3f^6+x^2f^5-4x^2f^4+2xf^3+(x+4)f^2-11f+7 = 0.

A256334 Number of C&C Family matchings on n edges.

Original entry on oeis.org

1, 1, 3, 12, 51, 227, 1052, 5030, 24634, 122950, 623140, 3198502, 16593124, 86864578, 458294970, 2434421685, 13008748377, 69882215729, 377172620330, 2044303447067, 11122504636031, 60723579401396, 332564474286299, 1826591420755058, 10058928726906713, 55528582177881182, 307224615377125853, 1703330011411361882, 9461963582991098963, 52655804456941167376, 293523046295844013225

Offset: 0

Aziza Jefferson, Mar 25 2015

nonn

The C&C Family of matchings is the family of matchings formed by first vertex insertions into the hairpin (except beneath both edges) or single edge (as long as the inserted edge does not have an outer edge connecting the first and last vertex), then edge inflations by ladders of the original single edge or hairpin.

			a(3)=12 because of the 15 matchings on 3 edges, three do not lie in the C&C Family. In canonical sequence form the missing matchings are given by 121323, 123123, and 123312.

S. Cao and S.-J. Chen, Predicting structures and stabilities for H-type pseudo knots with inter helix loops, RNA 15 (2009), 696-706.
Aziza Jefferson, The Substitution Decomposition of Matchings and RNA Secondary Structures, PhD Thesis, University of Florida, 2015.
C. Saule, M. Régnier, J.-M. Steyaert, and A. Denise, Counting RNA pseudoknotted structures, J. Comput. Biol. 18(10), (2011), 1339-1351.

f := RootOf(x^2*_Z^3 + x*(1-x)^2*_Z^2 - (1-x)^2*_Z + (1-x)^2);
series(f, x=0, 30);

f[x_] = Root[x^2 #^3 + x(1-x)^2 #^2 - (1-x)^2 # + (1-x)^2&, 1];
CoefficientList[f[x] + O[x]^31, x] (* Jean-François Alcover, Oct 06 2019 *)

G.f. f satisfies f = 1 + x*f^2 + (x^2*f^3)/(1-x)^2.

a(0)=1 prepended by Alois P. Heinz, Jul 14 2017

A256333 Number of R&G Family matchings on n edges.

Original entry on oeis.org

1, 3, 13, 61, 301, 1552, 8277, 45284, 252753, 1433633, 8239993, 47887467, 280927846, 1661387046, 9894376821, 59288650788, 357198545904, 2162437157263, 13147835385477, 80251977589719, 491573099486143, 3020738578507674, 18617035563669489, 115046892012376542, 712710925868858139, 4425312432316379040, 27535525144298975942, 171670784266383750322, 1072246008621559982926, 6708644077265798380125

Offset: 1

Aziza Jefferson, Mar 25 2015

nonn

The R&G Family of matchings is the family of matchings formed by first vertex insertions into the hairpin or single edge (as long as the inserted edge does not have an outer edge connecting the first and last vertex), then edge inflations by ladders of the original single edge or hairpin.

			a(3)=13 because of the 15 matchings on 3 edges, two do not lie in the R&G Family. In canonical sequence form the missing matchings are given by 121323 and 123123. a(4)= 61 out of the 105 matchings on 4 edges, one such matching which does not lie in the R&G Family is given by 12234314.

Aziza Jefferson, The Substitution Decomposition of Matchings and RNA Secondary Structures, PhD Thesis, University of Florida, 2015.
J. Reeder and R. Giegerich, Design, implementation and evaluation of a practical pseudo knot folding algorithm based on thermodynamics, BMC Bioinform. 5 (2004), Article #104.
C. Saule, M. Régnier, J.-M. Steyaert, and A. Denise, Counting RNA pseudoknotted structures, J. Comput. Biol. 18(10), (2011), 1339-1351.

f := RootOf(x^2*_Z^4 + x*(1-x)*(_Z-x*_Z)*_Z - (1-x)^2*_Z + (1-x)^2);
series(f, x=0, 30);

G.f. f satisfies x^2f^4 + x(1-x)^2f^2 - (1-x)^2f + (1-x)^2.

A256332 Number of D&P Family matchings on n edges.

Original entry on oeis.org

1, 3, 13, 65, 351, 1994, 11747, 71117, 439765, 2765775, 17636697, 113766694, 741032618, 4867177299, 32199559769, 214369107989, 1435126789097, 9655274425496, 65246685081291, 442668997422749, 3014127038713923, 20590331364902095, 141078438156193677, 969270926188235574, 6676082724399618966, 46089922748156948822, 318876966533117953114, 2210580887889464667057, 15353093117180070481879, 106816339860746421126519

Offset: 1

Aziza Jefferson, Mar 25 2015

nonn

			a(3)=13 because of the 15 matchings on 3 edges, two do not lie in the D&P Family. In canonical sequence form, the missing matchings are given by 121323 and 123123.

A. Condon, B. Davy, B. Rastegari, S. Zhao and F. Tarrant, RNA pseudoknotted structures, Theoret. Comput. Sci. 320(1), (2004), 35-50.
R. M. Dirks and N. A. Pierce, A partition function algorithm for nucleic acid secondary structure including pseudoknots, J. Compute. Chem. 24 (2003), 1664-1677.
Aziza Jefferson, The Substitution Decomposition of Matchings and RNA Secondary Structures, PhD Thesis, University of Florida, 2015.
C. Saule, M. Régnier, J.-M. Steyaert, and A. Denise, Counting RNA pseudoknotted structures, J. Comput. Biol. 18(10), (2011), 1339-1351.

f := RootOf(x^3*_Z^6-x^2*_Z^5+2*x*_Z^3-x*_Z^2-_Z+1);
series(f, x=0, 30);

G.f. f satisfies x^3f^6-x^2f^5+2xf^3-xf^2-f+1=0.

A256331 Number of Largest Hairpin Family matchings on n edges.

Original entry on oeis.org

1, 3, 14, 81, 527, 3684, 27022, 205149, 1598303, 12705939, 102653652, 840419676, 6956988612, 58132229976, 489673597926, 4153635860373, 35449185841679, 304179698619129, 2622657870000646, 22710277017073785, 197418128701387895

Offset: 1

Aziza Jefferson, Mar 25 2015

nonn

The Largest Hairpin Family of matchings is the largest family of matchings formed by repeated edge inflations and vertex insertions into the single edge and the hairpin.

			a(3) = 14 because of the 15 matchings on 3 edges, only 1 does not lie in the Largest Hairpin Family. In canonical sequence form, the missing matching is given by 121323.

Aziza Jefferson, The Substitution Decomposition of Matchings and RNA Secondary Structures, PhD Thesis, University of Florida, 2015.

f := RootOf(_Z^3*x-2*_Z^2*x-2*_Z^2+5*_Z-3, 1);
series(f, x=0, 30);

G.f. f satisfies x*f^3 - (2*x+2)*f^2 + 5*f - 3 = 0.

A256330 Number of H&S Family matchings on n edges.

Original entry on oeis.org

1, 3, 14, 84, 592, 4659, 39699, 359004, 3399164, 33378417, 337584612, 3498553682, 37006524557, 398312230440, 4351822041763, 48169486233388, 539303075161814, 6099303431601708, 69604032964928589, 800737747350839332, 9279033826462097649, 108236883894562489628

Offset: 1

Aziza Jefferson, Mar 25 2015

nonn

The H&S Family of matchings is the family of matchings that can be drawn in the plane without crossings.
Jay Pantone has computed the first 1500 terms and has a conjectured g.f. - N. J. A. Sloane, Oct 06 2016
Consider the graph whose vertices are the arcs of a matching, where two vertices are connected if the corresponding arcs cannot be drawn on the same side without crossing. Matchings, where the graph obtained this way is connected, are in bijection with 3-edge-connected rooted cubic maps with 2n nodes and a distinguished Hamiltonian cycle (cf. A000264). - Ludovic Schwob, Jun 17 2025

			a(5)= 592; in canonical sequence form the two 3-noncrossing matchings it does not include are 1231435425 and 1234254153.

Ludovic Schwob, Table of n, a(n) for n = 1..200
Michael Albert and Mireille Bousquet-Mélou, Permutations sortable by two stacks in parallel and quarter plane walks, European Journal of Combinatorics 43 (2015): 131-164. Also arXiv:1312.4487 [math.CO], 2013-2014.
C. Haslinger and P. F. Stadler, RNA structures with pseudo-notes: Graph-theoretical, combinatorial, and statistical properties, Bulletin of Mathematical Biology 61 (1999), 437-467.
Aziza Jefferson, The Substitution Decomposition of Matchings and RNA Secondary Structures, PhD Thesis, University of Florida, 2015.
Jay Pantone, Approximate Asymptotic Analysis of Combinatorial Sequences, Experimental Math Seminar, Rutgers University, Oct 06 2016.

Cf. A000264.

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User: Aziza Jefferson

A256338 Number of A&U Family matchings on n edges.

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A256337 Number of R&E Family matchings on n edges.

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A256336 Number of Largest Chain Ladder Family (LCLF) matchings on n edges.

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A256335 Number of Largest Chain Family matchings on n edges.

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A256334 Number of C&C Family matchings on n edges.

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A256333 Number of R&G Family matchings on n edges.

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A256332 Number of D&P Family matchings on n edges.

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A256331 Number of Largest Hairpin Family matchings on n edges.

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