Donovan Young - cp's OEIS frontend

A375505 Number of crystallized linear chord diagrams on n chords.

1, 2, 6, 25, 136, 923, 7557, 72767, 807896, 10180274, 143741731, 2250285510, 38715864581, 726596076239, 14780041925011, 324070919795226, 7622475922806634, 191515981769983447, 5120787153821434468, 145222986971201544125, 4355043425181710241819, 137728970544635824065325

Offset: 1

Donovan Young, Aug 23 2024

nonn

In a linear chord diagram a "bubble" is defined as a set of consecutive vertices such that no two adjacent vertices are joined by a chord, i.e., "short" chords are not allowed. A bubble is therefore bounded externally either by short chords, or by the ends of the diagram. In a crystallized diagram, all chords are either short, or "bridge" two distinct bubbles, i.e., they have one vertex in one bubble, and the other vertex in a separate bubble. a(n) is the total number of such diagrams built from n chords.

			For n = 3, let the vertices of the linear chord diagram be A,B,C,D,E,F. There are two diagrams with a single short chord: (AF)(BE)(CD) and (AE)(BF)(CD). There are three diagrams with two short chords: (AB)(CF)(DE), (AD)(BC)(EF), and (AF)(BC)(DE). Finally, there is one diagram with all three chords short: (AB)(CD)(EF). In total, there is therefore a(3) = 6 crystallized diagrams.

Donovan Young, Bubbles in Linear Chord Diagrams: Bridges and Crystallized Diagrams, arXiv:2408.17232 [math.CO], 2024. See p. 18.

Row sums of triangle A375504.

A375504 Triangle read by rows: T(n,k) is the number of crystallized linear chord diagrams on n chords with k short chords.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 6, 12, 6, 1, 24, 62, 39, 10, 1, 120, 396, 296, 95, 15, 1, 720, 3024, 2616, 980, 195, 21, 1, 5040, 26928, 26568, 11240, 2605, 357, 28, 1, 40320, 274320, 305892, 143464, 37290, 5971, 602, 36, 1, 362880, 3149280, 3945024, 2027460, 578514, 103824, 12292, 954, 45, 1

Offset: 0

Donovan Young, Aug 18 2024

In a linear chord diagram a "bubble" is defined as a set of consecutive vertices such that no two adjacent vertices are joined by a chord, i.e., "short" chords are not allowed. A bubble is therefore bounded externally either by short chords, or by the ends of the diagram. In a crystallized diagram, all chords are either short, or "bridge" two distinct bubbles, i.e., they have one vertex in one bubble, and the other vertex in a separate bubble. T(n,k) is the number of such crystallized diagrams built from n > 0 chords, exactly k > 0 of which are short.

			Triangle begins:
    1;
    1,   1;
    2,   3,   1;
    6,  12,   6,  1;
   24,  62,  39, 10,  1;
  120, 396, 296, 95, 15, 1;
  ...
For n = 3, let the vertices of the linear chord diagram be A,B,C,D,E,F. There are two diagrams with a single short chord: (AF)(BE)(CD) and (AE)(BF)(CD), and so T(3,1) = 2. There are three diagrams with two short chords: (AB)(CF)(DE), (AD)(BC)(EF), and (AF)(BC)(DE), and so T(3,2) = 3. Finally, there is one diagram with all three chords short: (AB)(CD)(EF), and so T(3,3)=1.

Donovan Young, Bubbles in Linear Chord Diagrams: Bridges and Crystallized Diagrams, arXiv:2408.17232 [math.CO], 2024.

Row sums give A375505.
First column gives A000142.
The second diagonal is A000217.

F[n_]:=Sum[Factorial2[2*i-1]*x^i,{i,0,n}];
T[n_,k_]:=Sum[(-1)^(n-k-l)*Factorial2[2*l-1]*Binomial[2*n-k,2*l]*Coefficient[F[n]^(k+1),x,n-k-l],{l,0,n-k}];

A367000 Triangle read by rows: T(n,k) is the total number of bubbles of size k found in linear chord diagrams on 2n vertices.

Original entry on oeis.org

0, 0, 2, 0, 0, 1, 8, 4, 2, 2, 0, 5, 42, 30, 20, 15, 12, 10, 0, 36, 300, 240, 186, 147, 120, 99, 82, 72, 0, 329, 2730, 2310, 1920, 1605, 1356, 1155, 988, 848, 730, 658, 0, 3655, 30240, 26460, 22890, 19845, 17280, 15105, 13242, 11634, 10240, 9027, 7968, 7310, 0, 47844

Offset: 0

Donovan Young, Oct 31 2023

A bubble is defined as a set of consecutive vertices such that no two adjacent vertices are joined by a chord, i.e., "short" chords are not allowed. A bubble is therefore bounded externally either by short chords, or by the ends of the diagram. T(n,k) counts the total number of bubbles consisting of k > 0 vertices, counted across all linear chord diagrams on 2n > 0 vertices.

			The first few rows of T(n,k) are:
   0,   0;
   2,   0,   0,   1;
   8,   4,   2,   2,   0,   5;
  42,  30,  20,  15,  12,  10,   0,  36;
For n = 2, let the four vertices be A, B, C, D. The diagram consisting of the chords (A,B) and (C,D) has no bubbles. The diagram consisting of the chords (A,D) and (B,C) has two bubbles of size 1: The vertex A is one bubble and the vertex D is the other. The diagram consisting of the chords (A,C) and (B,D) is itself a bubble of size 4. Hence T(2,1) = 2 and T(2,4) = 1.

Donovan Young, Counting Bubbles in Linear Chord Diagrams, arXiv:2311.01569 [math.CO], 2023.
Donovan Young, Bubbles in Linear Chord Diagrams: Bridges and Crystallized Diagrams, arXiv:2408.17232 [math.CO], 2024.

The last entry in each row forms A278990. See also A079267.

N=2*n;
G=0; for(j=0,j=N/2, G=G+taylor((1/((1 + w*(-1 + w*y^2))^2))*((((w^2*y^2)/(2*(1 + w^2*y^2)^2))^j*(2*j)!/j!* (-1 + w)^2*(-1 + w*y^2)^2)/(1 + w^2*y^2) - ((y^2)/2)^j/j!*w*y^2*((-2 + 2*w + (3 -4*w)*w*y^2 + (w + 2*(-1 + w)*w^2)*y^4 + w^3*y^6 )*(2*j)!+(-y^4 + w*y^4+ w*y^6 - 2*w^2*y^6 + w^3*y^8 )*(2*j+2)!)),y,N+1); );
Tn=vector(N,x,0);
for(k=1,k=N,Tn[k]=polcoeff(polcoeff(G,N,y),k,w););

G.f.: Sum_{j=0..n} (1/(1 + w*(-1 + w*y^2))^2)*((((w^2*y^2)/(2*(1 + w^2*y^2)^2))^j*((2*j)!/j!)* (-1 + w)^2*(-1 + w*y^2)^2)/(1 + w^2*y^2) - ((y^2)/2)^j*(w*y^2/j!)*((-2 + 2*w + (3 - 4*w)*w*y^2 + (w + 2*(-1 + w)*w^2)*y^4 + w^3*y^6)*(2*j)! + (-y^4 + w*y^4 + w*y^6 - 2*w^2*y^6 + w^3*y^8)*(2*j+2)!)).

A361378 Number of musical scales in n tone equal temperament respecting the property that alternate notes are 3 or 4 semitones apart.

Original entry on oeis.org

0, 1, 2, 3, 3, 3, 8, 8, 12, 16, 25, 33, 45, 66, 91, 128, 177, 252, 351, 491, 689, 966, 1354, 1894, 2658, 3723, 5217, 7309, 10244, 14355, 20112, 28185, 39494, 55343, 77547, 108667, 152272, 213372, 298992, 418968, 587089, 822665, 1152777, 1615350

Offset: 1

Donovan Young, Mar 09 2023

If you take any three consecutive notes in the scales counted by a(n) (with cyclic identification) then the distance between the first and third is either 3 or 4 semitones. a(n) is also the number of subsets of Z/nZ that 1) contain 0; 2) contain no subset of the form {x,x+1,x+2}; 3) have no superset satisfying property 2).

			For n=4 there are four notes, call them 0, 1, 2, and 3. The scales are 01, 02, and 03 and so a(4)=3.

Martin Epstein, mastodon.xyz thread, Feb 23 2023.
Donovan Young, The number of scales with the natural thirds property.
Index entries for linear recurrences with constant coefficients, signature (0,1,1,1,0,-1).

LinearRecurrence[{0,1,1,1,0,-1},{0,1,2,3,3,3},100]

a(n) = a(n-2) + a(n-3) + a(n-4) - a(n-6).

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

A336600 Triangle read by rows: T(n,k) is the number of linear chord diagrams on 2n vertices with one marked chord such that exactly k of the remaining n-1 chords contain the marked chord.

Original entry on oeis.org

1, 5, 1, 32, 11, 2, 260, 116, 38, 6, 2589, 1344, 594, 174, 24, 30669, 17529, 9294, 3774, 984, 120, 422232, 257487, 153852, 76782, 28272, 6600, 720, 6633360, 4234320, 2746260, 1576980, 726480, 242640, 51120, 5040, 117193185, 77358600, 53170380, 33718500, 18171360, 7693200, 2340720, 448560, 40320

Offset: 1

Donovan Young, Jul 30 2020

			Triangle begins:
     1;
     5,    1;
    32,   11,    2;
   260,  116,   38,   6;
  2589, 1344,  594, 174, 24;
...
For n = 2 and k = 1, let the four vertices be {1,2,3,4}. The marked chord can only be (2,3) and it is contained by one other chord, namely (1,4), hence T(2,1) = 1.

Donovan Young, Table of n, a(n) for n = 1..9870
Donovan Young, A critical quartet for queuing couples, arXiv:2007.13868 [math.CO], 2020.

Row sums are n*A001147(n) for n > 0.
Leading diagonal is A000142(n-1) for n > 0.
Sub-leading diagonal is A001344(n-2) for n > 1.
Cf. A336598, A336599, A336601.

CoefficientList[Normal[Series[Log[(1-x*(1+y))/(1-2*x)]/(1-y)/Sqrt[1-2*x],{x,0,10}]]/.{x^n_.->x^n*n!},{x,y}]

E.g.f.: log((1 - x*(1 + y))/(1 - 2*x))/(1 - y)/sqrt(1 - 2*x).

A336601 Triangle read by rows: T(n,k) is the number of linear chord diagrams on 2n vertices with one marked chord such that exactly k of the remaining n-1 chords are excluded by (i.e., are outside and do not contain) the marked chord.

Original entry on oeis.org

1, 4, 2, 22, 16, 7, 160, 136, 88, 36, 1464, 1344, 1044, 624, 249, 16224, 15504, 13344, 9624, 5484, 2190, 211632, 206592, 188952, 152832, 104322, 58080, 23535, 3179520, 3139200, 2977920, 2594880, 1990080, 1309680, 725040, 299880, 54092160, 53729280, 52096320, 47681280, 39652560, 29174400, 18809640, 10473120, 4426065

Offset: 1

Donovan Young, Jul 31 2020

			Triangle begins:
     1;
     4,    2;
    22,   16,    7;
   160,  136,   88,  36;
  1464, 1344, 1044, 624, 249;
...
For n = 2 and k = 1, let the four vertices be {1,2,3,4}. The marked chord can either be (1,2) and it excludes one other chord, namely (3,4), or vice-versa, hence T(2,1) = 2.

Donovan Young, Table of n, a(n) for n = 1..9870
Donovan Young, A critical quartet for queuing couples, arXiv:2007.13868 [math.CO], 2020.

Row sums are n*A001147(n) for n > 0.
The first column is A087547(n) for n > 0.
Leading diagonal is A034430(n-1) for n > 0.
Cf. A336598, A336599, A336600.

CoefficientList[Normal[Series[1/(1-y)/Sqrt[1-2*x]*ArcTan[(x*(1-y))/Sqrt[(1-2*x)]/Sqrt[1-2*y*x]],{x,0,10}]]/.{x^n_.->x^n*n!},{x,y}]

E.g.f.: arctan(x*(1 - y)/sqrt((1 - 2*x)*(1 - 2*x*y)))/(1 - y)/sqrt(1 - 2*x).

A336598 Triangle read by rows: T(n,k) is the number of linear chord diagrams on 2n vertices with one marked chord such that exactly k of the remaining n-1 chords cross the marked chord.

Original entry on oeis.org

1, 4, 2, 21, 18, 6, 144, 156, 96, 24, 1245, 1500, 1260, 600, 120, 13140, 16470, 16560, 11160, 4320, 720, 164745, 207270, 231210, 194040, 108360, 35280, 5040, 2399040, 2976120, 3507840, 3402000, 2419200, 1149120, 322560, 40320

Offset: 1

Donovan Young, Jul 29 2020

			Triangle begins:
     1;
     4,    2;
    21,   18,    6;
   144,  156,   96,  24;
  1245, 1500, 1260, 600, 120;
...
For n = 2 and k = 1, let the four vertices be {1,2,3,4}. The marked chord can be either (1,3), and so crossed once by (2,4), or (2,4), and so crossed once by (1,3). Hence T(2,1) = 2.

Donovan Young, Table of n, a(n) for n = 1..9870 (Rows 1..140).
Donovan Young, A critical quartet for queuing couples, arXiv:2007.13868 [math.CO], 2020.

Row sums are n*A001147(n) for n > 0.
First column is A233481(n) for n > 0.
Leading diagonal is A000142(n) for n > 0.
Sub-leading diagonal is n*A000142(n) for n > 1.
Cf. A336599, A336600, A336601.

CoefficientList[Normal[Series[x/Sqrt[1-2*x]/(1-x(1+y)),{x,0,10}]]/.{x^n_.->x^n*n!},{x,y}]

T(n)={[Vecrev(p) | p<-Vec(serlaplace(x/sqrt(1 - 2*x + O(x^n))/(1 - x*(1 + y))))]}
{ my(A=T(8)); for(n=1, #A, print(A[n])) } \\ Andrew Howroyd, Jul 29 2020

T(n,k) = n*T(n-1,k) + n*T(n-1,k-1), with T(n,0) = A233481(n) for n > 0.

E.g.f.: x/sqrt(1 - 2*x)/(1 - x*(1 + y)).

A336599 Triangle read by rows: T(n,k) is the number of linear chord diagrams on 2n vertices with one marked chord such that exactly k of the remaining n-1 chords are contained within the marked chord.

Original entry on oeis.org

1, 5, 1, 33, 9, 3, 279, 87, 39, 15, 2895, 975, 495, 255, 105, 35685, 12645, 6885, 4005, 2205, 945, 509985, 187425, 106785, 66465, 41265, 23625, 10395, 8294895, 3133935, 1843695, 1198575, 795375, 513135, 301455, 135135, 151335135, 58437855, 35213535, 23601375, 16343775, 11263455, 7453215, 4459455, 2027025

Offset: 1

Donovan Young, Jul 29 2020

			Triangle begins:
     1;
     5,    1;
    33,    9,    3;
   279,   87,   39,  15;
  2895,  975,  495, 255, 105;
...
For n = 2 and k = 1, let the four vertices be {1,2,3,4}. The marked chord can only be (1,4) and it contains one other chord, namely (2,3), hence T(2,1) = 1.

Donovan Young, Table of n, a(n) for n = 1..9870
Donovan Young, A critical quartet for queuing couples, arXiv:2007.13868 [math.CO], 2020.

Row sums are n*A001147(n) for n > 0.
Leading diagonal is A001147(n-1) for n > 0.
The first column is A129890(n-1) for n > 0.
The second column is A035101(n+1) for n > 0.
Cf. A336598, A336600, A336601.

CoefficientList[Normal[Series[(Sqrt[1-2*y*x]-Sqrt[1-2*x])/(1-2*x)/(1-y),{x,0,10}]]/.{x^n_.->x^n*n!},{x,y}]

E.g.f.: (sqrt(1 - 2*y*x) - sqrt(1 - 2*x))/(1 - 2*x)/(1 - y).

A334062 Triangle read by rows: T(n,k) is the number of non-crossing set partitions of {1..4n} into n sets of 4 with k of the sets being a contiguous set of elements.

Original entry on oeis.org

1, 3, 1, 9, 12, 1, 27, 81, 31, 1, 81, 432, 390, 65, 1, 243, 2025, 3330, 1365, 120, 1, 729, 8748, 22815, 17415, 3909, 203, 1, 2187, 35721, 135513, 166320, 70938, 9730, 322, 1, 6561, 139968, 728028, 1312038, 911358, 242004, 21816, 486, 1, 19683, 531441, 3630420, 9032310, 9294264, 4067658, 722316, 45090, 705, 1

Offset: 1

Donovan Young, May 28 2020

T(n,k) is also the number of non-crossing configurations with exactly k polyomino matchings in a generalized game of memory played on the path of length 4n, see [Young].
For the case of partitions of {1..3n} into sets of 3, see A091320.
For the case of partitions of {1..2n} into sets of 2, see A001263.

			Triangle starts:
    1;
    3,    1;
    9,   12,    1;
   27,   81,   31,    1;
   81,  432,  390,   65,   1;
  243, 2025, 3330, 1365, 120, 1;
  ...
For n=2 and k=1 the configurations are (1,6,7,8),(2,3,4,5), (1,2,7,8),(3,4,5,6), and (1,2,3,8),(4,5,6,7); hence T(2,1) = 3.

Donovan Young, Linear k-Chord Diagrams, arXiv:2004.06921 [math.CO], 2020.

Row sums are A002293.
Column 2 is A069996.
Cf. A001263, A091320, A334063.

G.f.: G(t, z) satisfies z*G^4 - (1 + z - t*z)*G + 1 = 0.

A334063 Triangle read by rows: T(n,k) is the number of non-crossing set partitions of {1..5n} into n sets of 5 with k of the sets being a contiguous set of elements.

Original entry on oeis.org

1, 4, 1, 16, 18, 1, 64, 168, 52, 1, 256, 1216, 936, 121, 1, 1024, 7680, 11040, 3760, 246, 1, 4096, 44544, 103040, 67480, 12264, 455, 1, 16384, 243712, 827904, 888160, 318976, 34524, 784, 1, 65536, 1277952, 5992448, 9554944, 5716704, 1254512, 86980, 1278, 1

Offset: 1

Donovan Young, May 28 2020

T(n,k) is also the number of non-crossing configurations with exactly k polyomino matchings in a generalized game of memory played on the path of length 5n, see [Young].
For the case of partitions of {1..4n} into sets of 4, see A334062.
For the case of partitions of {1..3n} into sets of 3, see A091320.
For the case of partitions of {1..2n} into sets of 2, see A001263.

			Triangle starts:
     1;
     4,    1;
    16,   18,     1;
    64,  168,    52,    1;
   256, 1216,   936,  121,   1;
  1024, 7680, 11040, 3760, 246,  1;
  ...
For n = 2 and k = 1 the configurations are (1,7,8,9,10), (2,3,4,5,6), (1,2,8,9,10),(3,4,5,6,7), (1,2,3,9,10), (4,5,6,7,8) and (1,2,3,4,10), (5,6,7,8,9); hence T(2,1) = 4.

Donovan Young, Linear k-Chord Diagrams, arXiv:2004.06921 [math.CO], 2020.

Row sums are A002294.

Cf. A001263, A091320, A334062.

G.f.: G(t, z) satisfies z*G^5 - (1 + z - t*z)*G + 1 = 0.

cp's OEIS Frontend

User: Donovan Young

A375505 Number of crystallized linear chord diagrams on n chords.

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A375504 Triangle read by rows: T(n,k) is the number of crystallized linear chord diagrams on n chords with k short chords.

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A367000 Triangle read by rows: T(n,k) is the total number of bubbles of size k found in linear chord diagrams on 2n vertices.

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A361378 Number of musical scales in n tone equal temperament respecting the property that alternate notes are 3 or 4 semitones apart.

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A336600 Triangle read by rows: T(n,k) is the number of linear chord diagrams on 2n vertices with one marked chord such that exactly k of the remaining n-1 chords contain the marked chord.

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A336601 Triangle read by rows: T(n,k) is the number of linear chord diagrams on 2n vertices with one marked chord such that exactly k of the remaining n-1 chords are excluded by (i.e., are outside and do not contain) the marked chord.

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A336598 Triangle read by rows: T(n,k) is the number of linear chord diagrams on 2n vertices with one marked chord such that exactly k of the remaining n-1 chords cross the marked chord.

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A336599 Triangle read by rows: T(n,k) is the number of linear chord diagrams on 2n vertices with one marked chord such that exactly k of the remaining n-1 chords are contained within the marked chord.

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