Lily Yen - cp's OEIS frontend

A225031 Non-crossing, non-nesting, 5-colored set partitions.

1, 6, 41, 321, 2846, 27961, 297681, 3371646, 40065361, 494281201, 6279901766, 81649478161, 1080910639201, 14511820543126, 196956264035481, 2695543342918241, 37127978351861646, 513895401953712521, 7139331902125917361, 99462520534916445006, 1388616983941077336321

Offset: 0

Lily Yen, Apr 25 2013

			For n=2, a(2)=41 is the number of non-crossing, non-nesting set partitions on 3 elements with 5 possible arc colors.

Lily Yen, Table of n, a(n) for n = 0..99
Eric Marberg, Crossings and nestings in colored set partitions, arXiv preprint arXiv:1203.5738 [math.CO], 2012-2013.
Lily Yen, Crossings and Nestings for Arc-Coloured Permutations, and Arc-coloured permutations, PSAC 2013, Paris, France, June 24-28, Proc. DMTCS (2013) 743-754.
Lily Yen, Crossings and Nestings for Arc-Coloured Permutations and Automation, Electronic Journal of Combinatorics, 22(1) (2015), #P1.14.
Index entries for linear recurrences with constant coefficients, signature (41,-638,4701,-16398,21721,-1).

LinearRecurrence[{41, -638, 4701, -16398, 21721, -1}, {1, 6, 41, 321, 2846, 27961}, 21] (* Jean-François Alcover, Jul 22 2018 *)

Vec((1 -35*x +433*x^2 -2233*x^3 +4035*x^4 -x^5) / (1 -41*x +638*x^2 -4701*x^3 +16398*x^4 -21721*x^5 +x^6) + O(x^66)) \\ Joerg Arndt, Apr 27 2013

G.f.: (1 -35*x +433*x^2 -2233*x^3 +4035*x^4 -x^5) / (1 -41*x +638*x^2 -4701*x^3 +16398*x^4 -21721*x^5 +x^6).

A224993 Non-crossing, non-nesting, 4-colored permutations on {1,2,...,n}.

Original entry on oeis.org

1, 4, 32, 352, 4736, 72832, 1226240, 21948928, 409192448, 7833143296, 152494727168, 3000118779904, 59406517698560, 1180988766453760, 23534128521936896, 469655122210324480, 9380774946206646272, 187467580232576794624, 3747576648059504820224

Offset: 0

Lily Yen, Apr 24 2013

A225029-A225033 are sequences counting non-crossing, non-nesting, r-colored set partitions for r=3..7. Set partitions only have upper arcs, whereas permutations have upper and lower arcs in their annotated arc diagram representations.

			For n=3, a(3)=352, the number of ways to color arcs of a permutation on 3 elements in 4 colors so that arcs of the same color do not cross nor nest.

Lily Yen, Table of n, a(n) for n = 0..99
Lily Yen, Crossings and Nestings for Arc-Coloured Permutations and Arc-coloured permutations, PSAC 2013, Paris, France, June 24-28, Proc. DMTCS (2013) 743-754.
Index entries for linear recurrences with constant coefficients, signature (40,-508,2304,-2880).

Join[{1}, LinearRecurrence[{40, -508, 2304, -2880}, {4, 32, 352, 4736}, 20]] (* Jean-François Alcover, Jul 22 2018 *)

Vec((1-36*x+380*x^2-1200*x^3+576*x^4)/((1-2*x)*(1-6*x)*(1-12*x)*(1-20*x)) +O(x^66)) \\ Joerg Arndt, Apr 24 2013

G.f.: (1-36*x+380*x^2-1200*x^3+576*x^4)/((1-2*x)*(1-6*x)*(1-12*x)*(1-20*x)).

a(n) = 2^(n-1)*(20*3^n+7*6^n+10^n+28)/35 for n>0, a(0)=1. [Bruno Berselli, Apr 26 2013]

A225033 Non-crossing, non-nesting, 7-colored set partitions.

Original entry on oeis.org

1, 8, 71, 715, 8212, 106205, 1514633, 23353828, 383455843, 6630981491, 119760987872, 2243989397161, 43378019032321, 861000869284928, 17476961860459151, 361541275664799595, 7599788958355060972, 161922899182197739685, 3489406153035009734633, 75917779229255330345308

Offset: 0

Lily Yen, Apr 25 2013

nonn

			For n=2, a(2)=71 is the number of non-crossing, non-nesting, 7-colored set partitions on 3 elements.

Muniru A Asiru, Table of n, a(n) for n = 0..150 (Terms n=0..99 from Lily Yen)
Eric Marberg, Crossings and nestings in colored set partitions, arXiv preprint arXiv:1203.5738 [math.CO], 2012-2013.
Lily Yen, Crossings and Nestings for Arc-Coloured Permutations, arXiv:1211.3472 [math.CO], 2012-2013; Formal Power Series and Algebraic Combinatorics Conference, June (2013), to appear.
Lily Yen, Crossings and Nestings for Arc-Coloured Permutations and Automation, Electronic Journal of Combinatorics, 22(1) (2015), #P1.14.

seq(coeff(series((1-84*x +2849*x^2 -49873*x^3 +474601*x^4 -2324333*x^5 +4567788*x^6 -x^7) / (1-92*x +3514*x^2 -72168*x^3 +860019*x^4 -5943768*x^5 +22055962*x^6 -33922100*x^7 +x^8),x,n+1), x, n), n = 0 .. 20); # Muniru A Asiru, Feb 22 2019

a = DifferenceRoot[Function[{a, n}, {a[n] - 33922100*a[n+1] + 22055962*a[n+2] - 5943768*a[n+3] + 860019*a[n+4] - 72168*a[n+5] + 3514*a[n+6] - 92*a[n+7] + a[n+8] == 0, a[0] == 1, a[1] == 8, a[2] == 71, a[3] == 715, a[4] == 8212, a[5] == 106205, a[6] == 1514633, a[7] == 23353828}]];
Table[a[n], {n, 0, 19}] (* Jean-François Alcover, Feb 22 2019 *)

G.f.: (1-84*x +2849*x^2 -49873*x^3 +474601*x^4 -2324333*x^5 +4567788*x^6 -x^7) / (1-92*x +3514*x^2 -72168*x^3 +860019*x^4 -5943768*x^5 +22055962*x^6 -33922100*x^7 +x^8).

A225032 Non-crossing, non-nesting, 6-colored set partitions.

Original entry on oeis.org

1, 7, 55, 493, 5029, 57379, 716443, 9604345, 136236937, 2022864031, 31180099711, 495615409957, 8079827006125, 134488017925243, 2276945808434659, 39088515241450609, 678651272689389073, 11890942901283331255, 209891714523969067207, 3727004974842239659741

Offset: 0

Lily Yen, Apr 25 2013

			For n=2, a(2)=55 is the number of non-crossing, non-nesting set partitions on 3 elements with 6 possible arc colors.

Lily Yen, Table of n, a(n) for n = 0..99
Eric Marberg, Crossings and nestings in colored set partitions, arXiv preprint arXiv:1203.5738 [math.CO], 2012-2013.
Lily Yen, Crossings and Nestings for Arc-Coloured Permutations, arXiv:1211.3472 [math.CO], 2012-2013 and Arc-coloured permutations, PSAC 2013, Paris, France, June 24-28, Proc. DMTCS (2013) 743-754.
Lily Yen, Crossings and Nestings for Arc-Coloured Permutations and Automation, Electronic Journal of Combinatorics, 22(1) (2015), #P1.14.
Index entries for linear recurrences with constant coefficients, signature (63,-1589,20515,-142915,509549,-727767,1).

Vec((1 - 56*x + 1203*x^2 - 12364*x^3 + 60675*x^4 - 113540*x^5 + x^6) / (1 - 63*x + 1589*x^2 - 20515*x^3 + 142915*x^4 - 509549*x^5 + 727767*x^6 - x^7) + O(x^40)) \\ Colin Barker, Jun 22 2019

G.f.: (1-56*x+1203*x^2-12364*x^3+60675*x^4-113540*x^5+x^6)/ (1-63*x+1589*x^2-20515*x^3+142915*x^4-509549*x^5+727767*x^6-x^7).

a(n) = 63*a(n-1) - 1589*a(n-2) + 20515*a(n-3) - 142915*a(n-4) + 509549*a(n-5) - 727767*a(n-6) + a(n-7) for n>6. - Colin Barker, Jun 22 2019

A225030 Non-crossing, non-nesting, 4-colored set partitions.

Original entry on oeis.org

1, 5, 29, 193, 1441, 11765, 102701, 941857, 8955937, 87439877, 870218525, 8780788513, 89476873345, 918150779957, 9467752541933, 97965021468865, 1016097175530433, 10556565963815045, 109802406545873309, 1143006276663287809, 11904902286515536609

Offset: 0

Lily Yen, Apr 25 2013

			For n=3, a(3)=193 is the number of non-crossing, non-nesting, 4-colored set partitions on 4 elements.

Lily Yen, Table of n, a(n) for n = 0..99
Eric Marberg, Crossings and nestings in colored set partitions, arXiv preprint arXiv:1203.5738 [math.CO], 2012-2013.
Lily Yen, Crossings and Nestings for Arc-Coloured Permutations, arXiv:1211.3472 [math.CO], 2012-2013 and Arc-coloured permutations, PSAC 2013, Paris, France, June 24-28, Proc. DMTCS (2013) 743-754
Lily Yen, Crossings and Nestings for Arc-Coloured Permutations and Automation, Electronic Journal of Combinatorics, 22(1) (2015), #P1.14.
Index entries for linear recurrences with constant coefficients, signature (25,-218,782,-973,1).

LinearRecurrence[{25, -218, 782, -973, 1}, {1, 5, 29, 193, 1441}, 25] (* Paolo Xausa, Feb 06 2024 *)

G.f.: (1-20*x+122*x^2-224*x^3+x^4)/(1-25*x +218*x^2-782*x^3+973*x^4-x^5).

A225029 Non-crossing, non-nesting, 3-colored set partitions.

Original entry on oeis.org

1, 4, 19, 103, 616, 3949, 26545, 184120, 1303135, 9341191, 67490044, 489978217, 3567727441, 26024391436, 190036459099, 1388593185079, 10150390743088, 74215146065461, 542704850311009, 3968914608295360, 29026988765886535, 212297824609934455, 1552734183515322436

Offset: 0

Lily Yen, Apr 24 2013

nonn

			a(3) = 103 is the number of non-crossing, non-nesting, 3-colored set partitions on {1,2,3,4}.

Muniru A Asiru, Table of n, a(n) for n = 0..1145 (first 100 terms from Lily Yen)
Eric Marberg, Crossings and nestings in colored set partitions, arXiv preprint arXiv:1203.5738 [math.CO], 2012-2013.
Lily Yen, Crossings and Nestings for Arc-Coloured Permutations, arXiv:1211.3472 [math.CO], 2012-2013; and Arc-coloured permutations, PSAC 2013, Paris, France, June 24-28, Proc. DMTCS (2013) 743-754
Lily Yen, Crossings and Nestings for Arc-Coloured Permutations and Automation, Electronic Journal of Combinatorics, 22(1) (2015), #P1.14.
Index entries for linear recurrences with constant coefficients, signature(14,-59,74,-1).

a:=[1,4,19,103];; for n in [5..25] do a[n]:=14*a[n-1]-59*a[n-2]+74*a[n-3]-a[n-4]; od; a; # Muniru A Asiru, Dec 18 2018

I:=[1,4,19,103]; [n le 4 select I[n] else 14*Self(n-1)-59*Self(n-2)+74*Self(n-3)-Self(n-4): n in [1..30]]; // Vincenzo Librandi, Dec 20 2018

seq(coeff(series((1-10*x+22*x^2-x^3)/(1-14*x+59*x^2-74*x^3+x^4),x,n+1), x, n), n = 0 .. 25); # Muniru A Asiru, Dec 18 2018

LinearRecurrence[{14, -59, 74, -1}, {1, 4, 19, 103}, 23] (* Jean-François Alcover, Dec 14 2018 *)
CoefficientList[Series[(1 - 10 x + 22 x^2 - x^3) / (1 - 14 x + 59 x^2 - 74 x^3 + x^4), {x, 0, 25}], x] (* Vincenzo Librandi, Dec 20 2018 *)

Vec((1-10*x+22*x^2-x^3)/(1-14*x+59*x^2-74*x^3+x^4)+O(x^66)) \\ Joerg Arndt, Apr 24 2013

G.f.: (1 - 10*x + 22*x^2 - x^3)/(1 - 14*x + 59*x^2 - 74*x^3 + x^4).

a(n) = 14*a(n-1) -59*a(n-2) +74*a(n-3) -a(n-4), with a(0) = 1, a(1) = 4, a(2) = 19 and a(3) = 103. - Muniru A Asiru, Dec 18 2018

A224992 Non-crossing, non-nesting, 3-colored permutations on {1,2,...,n}.

Original entry on oeis.org

1, 3, 18, 144, 1368, 14400, 160992, 1861632, 21919104, 260508672, 3110985216, 37241118720, 446349219840, 5352925446144, 64215514275840, 770468624990208, 9244918222258176, 110934787001942016, 1331192054033547264, 15974152308466384896, 191688913661984243712

Offset: 0

Lily Yen, Apr 24 2013

			For n=3, a(3)= 144, the number of ways to color arcs of a permutation on {1,2,3} in 3 colors such that the arcs neither cross nor nest.

Lily Yen, Table of n, a(n) for n = 0..99
Wei Chen, Enumeration of Set Partitions Refined by Crossing and Nesting Numbers, MS Thesis, Department of Mathematics. Simon Fraser University, Fall 2014. Table 5.2, r=3.
Lily Yen, Crossings and Nestings for Arc-Coloured Permutations and Arc-coloured permutations, PSAC 2013, Paris, France, June 24-28, Proc. DMTCS (2013) 743-754.
Index entries for linear recurrences with constant coefficients, signature (20,-108,144).

Join[{1}, LinearRecurrence[{20, -108, 144}, {3, 18, 144}, 20]] (* Jean-François Alcover, Jul 22 2018 *)

Vec((1-17*x+66*x^2-36*x^3)/((1-2*x)*(1-6*x)*(1-12*x))+O(x^66)) \\ Joerg Arndt, Apr 24 2013

G.f.: (1-17*x+66*x^2-36*x^3)/((1-2*x)*(1-6*x)*(1-12*x)).

a(n) = 9*2^n/20 +6^n/4 +12^n/20, n>0. - R. J. Mathar, Jun 11 2019

A219587 Noncrossing, nonnesting, 2-arc-colored permutations on the set {1..n} where lower arcs even of different colors do not cross.

Original entry on oeis.org

1, 2, 8, 40, 224, 1296, 7568, 44304, 259536, 1520656, 8910160, 52209040, 305919696, 1792542992, 10503446608, 61545189520, 360625475024, 2113093401616, 12381720203088, 72550979111824, 425114158957776, 2490966357221136, 14595875630354000, 85524874633320080

Offset: 0

Lily Yen, Nov 23 2012

The sequence is generated by a rational function, in particular, a quotient of two determinants.

			For n=4, the a(4) = 224 solutions are 24 permutations, 8 of which can be colored in 4 ways each, 8 in 8 ways each, and 8 in 16 ways each, thus resulting in 8 * (4+8+16) = 224.

Lily Yen, Table of n, a(n) for n = 0..1000
Lily Yen, Crossings and Nestings for Arc-Coloured Permutations, arXiv:1211.3472 [math.CO], 2012-2013.
Index entries for linear recurrences with constant coefficients, signature (7,-6,-4).

Vec((1 - 5*x) / (1 - 7*x + 6*x^2 + 4*x^3) + O(x^40)) \\ Colin Barker, Jun 22 2019

G.f.: (1 - 5*x)/(1 - 7*x + 6*x^2 + 4*x^3).

a(n) = 7*a(n-1) - 6*a(n-2) - 4*a(n-3) for n>2. - Colin Barker, Jun 22 2019

Name modified by Lily Yen, Apr 23 2013

A193938 3-nonnesting permutations.

Original entry on oeis.org

1, 2, 6, 24, 118, 675, 4333, 30464, 230615, 1856336, 15738672, 139509303, 1285276242, 12248071935, 120255584181, 1212503440774, 12519867688928, 132079067871313, 1420723274988736, 15554956521285848

Offset: 1

Lily Yen, Aug 09 2011

Also the number of 3-noncrossing permutations.

Sophie Burrill, Sergi Elizalde, Marni Mishna and Lily Yen, A generating tree approach to k-nonnesting partitions and permutations, arXiv preprint arXiv:1108.5615 [math.CO], 2011-2014.
S. Corteel, Crossings and alignments of permutations, Adv. Appl. Math 38 (2007) 149-163.

Cf. A193935, A193936, A193937.

A193937 6-nonnesting permutations.

Original entry on oeis.org

1, 2, 6, 24, 120, 720, 5040, 40320, 362880, 3628800, 39916680, 478991641, 6226516930, 87157924751, 1306945300264

Offset: 1

Lily Yen, Aug 09 2011

Also the number of 6-noncrossing permutations.

Sophie Burrill, Sergi Elizalde, Marni Mishna and Lily Yen, A generating tree approach to k-nonnesting partitions and permutations, arXiv preprint arXiv:1108.5615 [math.CO], 2011-2014.
S. Corteel, Crossings and alignments of permutations, Adv. Appl. Math 38 (2007) 149-163.

Cf. A193938, A193935, A193936.

cp's OEIS Frontend

User: Lily Yen

A225031 Non-crossing, non-nesting, 5-colored set partitions.

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A224993 Non-crossing, non-nesting, 4-colored permutations on {1,2,...,n}.

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A225033 Non-crossing, non-nesting, 7-colored set partitions.

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A225032 Non-crossing, non-nesting, 6-colored set partitions.

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A225030 Non-crossing, non-nesting, 4-colored set partitions.

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A225029 Non-crossing, non-nesting, 3-colored set partitions.

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A224992 Non-crossing, non-nesting, 3-colored permutations on {1,2,...,n}.

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A219587 Noncrossing, nonnesting, 2-arc-colored permutations on the set {1..n} where lower arcs even of different colors do not cross.

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