James East - cp's OEIS frontend

A340914 Square array, read by rows. For n,d >= 0, a(n,d) is the number of congruences of the d-twisted partition monoid of degree n.

2, 3, 3, 9, 7, 4, 12, 43, 14, 5, 16, 76, 136, 24, 6, 19, 134, 329, 334, 37, 7, 22, 188, 773, 1105, 696, 53, 8, 25, 251, 1281, 3456, 3100, 1294, 72, 9, 28, 323, 1969, 6754, 12806, 7608, 2213, 94, 10, 31, 404, 2864, 11930, 29413, 41054, 16842, 3551, 119, 11, 34, 494, 3993, 19578, 59547, 110312, 117273, 34353, 5419, 147, 12

Offset: 0

James East, Mar 07 2021

The d-twisted partition monoids P_{n,d} are defined in the East-Ruškuc paper cited below.

			Array begins:
=========================================================
n\d |  0   1    2     3      4      5       6       7 ...
----+----------------------------------------------------
  0 |  2   3    4     5      6      7       8       9 ...
  1 |  3   7   14    24     37     53      72      94 ...
  2 |  9  43  136   334    696   1294    2213    3551 ...
  3 | 12  76  329  1105   3100   7608   16842   34353 ...
  4 | 16 134  773  3456  12806  41054  117273  304889 ...
  5 | 19 188 1281  6754  29413 110312  366724 1103538 ...
  6 | 22 251 1969 11930  59547 255132  965409 3293916 ...
  7 | 25 323 2864 19578 110012 529298 2242845 8544569 ...
  ...

Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)
James East and Nik Ruškuc, Properties of congruences of twisted partition monoids and their lattices, arXiv:2010.09288 [math.RA], 2020-2021.

T(n,d) = {if(n<=3, if(n<=1, if(n<=0, if(n==0, d+2), (3*d^2+5*d+6)/2), if(n==2, (13*d^4+106*d^3+299*d^2+398*d+216)/24, (13*d^7+322*d^6+3262*d^5+17920*d^4+58597*d^3+115318*d^2+127128*d+60480)/5040)), binomial(3*n+d-4,3*n-5) + 8*binomial(3*n+d-1,3*n-1) +2*binomial(3*n+d-2,3*n-1) + 5*binomial(3*n+d-3,3*n-1) - 2*binomial(3*n+d-4,3*n-1))} \\ Andrew Howroyd, Jan 06 2024

a(0,d) = d+2,
a(1,d) = (3*d^2+5*d+6)/2,
a(2,d) = (13*d^4+106*d^3+299*d^2+398*d+216)/24,
a(3,d) = (13*d^7+322*d^6+3262*d^5+17920*d^4+58597*d^3+115318*d^2+127128*d+60480)/5040,
a(n,d) = binomial(3*n+d-4,3*n-5) + 8*binomial(3*n+d-1,3*n-1) + 2*binomial(3*n+d-2,3*n-1) + 5*binomial(3*n+d-3,3*n-1) - 2*binomial(3*n+d-4,3*n-1) for n >= 4.
For fixed d >= 0, a(n,d) is asymptotic to (3*n)^(d+1) / (d+1)!.
For fixed n >= 4, a(n,d) is asymptotic to 13*d^(3*n-1) / (3*n-1)!.
A rational generating function is given in the East-Ruškuc paper, and also polynomial expressions for a(n,d) with d fixed (and n >= 4).

A293822 Number of integer-sided pentagons having perimeter n, modulo rotations but not reflections.

Original entry on oeis.org

1, 1, 3, 6, 13, 21, 37, 51, 84, 108, 166, 203, 294, 350, 486, 566, 759, 867, 1133, 1276, 1631, 1815, 2275, 2509, 3094, 3386, 4116, 4473, 5372, 5804, 6896, 7412, 8721, 9333, 10887, 11606, 13433, 14269, 16401, 17367, 19836, 20944, 23782, 25047, 28290, 29726, 33410, 35030, 39195, 41015

Offset: 5

James East, Oct 16 2017

nonn

Rotations are counted only once, but reflections are considered different. For a polygon to be nondegenerate, the longest side must be shorter than the sum of the remaining sides (equivalently, shorter than n/2).

A formula is given in Section 6 of the East and Niles article.

			For example, there are 6 rotation-classes of perimeter-8 pentagons: 32111, 31211, 31121, 31112, 22211, 22121.  Note that 32111 and 31112 are reflections of each other, but these are not rotationally equivalent.

James East, Ron Niles, Integer polygons of given perimeter, arXiv:1710.11245 [math.CO], 2017.

Column k=5 of A293819.

Cf. A293820 (polygons), A293821 (quadrilaterals), A293823 (hexagons).

T[n_, k_] := DivisorSum[GCD[n, k], EulerPhi[#]*Binomial[n/#, k/#] &]/n - Binomial[Floor[n/2], k - 1];
a[n_] := T[n, 5];
Table[a[n], {n, 5, 60}] (* Jean-François Alcover, Jun 14 2018, after Andrew Howroyd and A293819 *)

G.f.: x^5*(1 + x - x^2 + 2*x^3 + 7*x^4 + 2*x^5 - 2*x^6 + x^8) / ((1 - x)^5*(1 + x)^4*(1 + x + x^2 + x^3 + x^4)) (conjectured). - Colin Barker, Nov 01 2017

A293823 Number of integer-sided hexagons having perimeter n, modulo rotations but not reflections.

Original entry on oeis.org

1, 1, 4, 10, 21, 41, 74, 126, 196, 314, 448, 672, 912, 1302, 1692, 2334, 2937, 3927, 4828, 6292, 7579, 9679, 11466, 14378, 16808, 20748, 23968, 29198, 33388, 40188, 45564, 54264, 61047, 72033, 80484, 94164, 104587, 121429, 134134, 154672, 170016, 194810, 213200, 242880, 264730, 300002

Offset: 6

James East, Oct 16 2017

nonn

Rotations are counted only once, but reflections are considered different. For a polygon to be nondegenerate, the longest side must be shorter than the sum of the remaining sides (equivalently, shorter than n/2).

A formula is given in Section 6 of the East and Niles article.

			For example, there are 10 rotation-classes of perimeter-9 hexagons: 411111, 321111, 312111, 311211, 311121, 311112, 222111, 221211, 221121, 212121. Note that 321111 and 311112 are reflections of each other, but these are not rotationally equivalent.

James East, Ron Niles, Integer polygons of given perimeter, arXiv:1710.11245 [math.CO], 2017.

Column k=6 of A293819.

Cf. A293820 (polygons), A293822 (pentagons).

T[n_, k_] := DivisorSum[GCD[n, k], EulerPhi[#]*Binomial[n/#, k/#] &]/n - Binomial[Floor[n/2], k - 1];
a[n_] := T[n, 6];
Table[a[n], {n, 6, 51}] (* Jean-François Alcover, Jan 29 2019, after Andrew Howroyd in A293819 *)

G.f.: x^6*(1 + x + 5*x^3 + 10*x^4 + 7*x^5 + 3*x^6 + 6*x^7 + 4*x^8 + 2*x^9) / ((1 - x)^6*(1 + x)^5*(1 - x + x^2)*(1 + x + x^2)^2) (conjectured). - Colin Barker, Nov 01 2017

A293821 Number of integer-sided quadrilaterals having perimeter n, modulo rotations but not reflections.

Original entry on oeis.org

1, 1, 2, 4, 6, 10, 12, 20, 23, 35, 38, 56, 60, 84, 88, 120, 125, 165, 170, 220, 226, 286, 292, 364, 371, 455, 462, 560, 568, 680, 688, 816, 825, 969, 978, 1140, 1150, 1330, 1340, 1540, 1551, 1771, 1782, 2024, 2036, 2300, 2312, 2600, 2613, 2925, 2938, 3276, 3290, 3654, 3668, 4060

Offset: 4

James East, Oct 16 2017

nonn

Rotations are counted only once, but reflections are considered different. For a polygon to be nondegenerate, the longest side must be shorter than the sum of the remaining sides (equivalently, shorter than n/2).

A formula is given in Section 6 of the East and Niles article.

			For example, there are 4 rotation-classes of perimeter-7 quadrilaterals: 3211, 3121, 3112, 2221. Note that 3211 and 3112 are reflections of each other, but these are not rotationally equivalent.

James East, Ron Niles, Integer polygons of given perimeter, arXiv:1710.11245 [math.CO], 2017.

Column k=4 of A293819.

Cf. A008742 (triangles), A293820 (polygons), A293822 (pentagons).

T[n_, k_] := DivisorSum[GCD[n, k], EulerPhi[#]*Binomial[n/#, k/#] &]/n - Binomial[Floor[n/2], k - 1];
a[n_] := T[n, 4];
Table[a[n], {n, 4, 59}] (* Jean-François Alcover, Jan 29 2019, after Andrew Howroyd in A293819 *)

Conjectures from Colin Barker, Nov 01 2017: (Start)
G.f.: x^3*(1 - x^2 + 2*x^3) / ((1 - x)^4*(1 + x)^3*(1 + x^2)).
a(n) = (1/96)*(-3*(-1 + (-1)^n + 4*i*(-i)^n - 4*i*i^n) + (7 - 15*(-1)^n)*n + 3*(-1 + (-1)^n)*n^2 + 2*n^3) where i=sqrt(-1).
(End)

A293820 Number of integer-sided polygons having perimeter n, modulo rotations but not reflections.

Original entry on oeis.org

1, 1, 3, 5, 11, 19, 43, 75, 155, 287, 567, 1053, 2063, 3859, 7455, 14089, 27083, 51463, 98855, 188697, 362675, 695155, 1338087, 2573235, 4962875, 9571195, 18496407, 35759799, 69240899, 134154259, 260235639, 505163055, 981575759, 1908619755, 3714304167, 7233118641, 14095779055

Offset: 3

James East, Oct 16 2017

nonn

Rotations are counted only once, but reflections are considered different. For a polygon to be nondegenerate, the longest side must be shorter than the sum of the remaining sides (equivalently, shorter than n/2). These are row sums of A293819.
A formula is given in Section 6 of the East and Niles article.
The same article shows that a(n) is asymptotic to 2^n / n.

			There are 11 polygons having perimeter 7: 2 triangles (331, 322), 4 quadrilaterals (3211, 3121, 3112, 2221), 3 pentagons (31111, 22111, 21211), 1 hexagon (211111) and 1 heptagon (1111111).

Andrew Howroyd, Table of n, a(n) for n = 3..200
James East, Ron Niles, Integer polygons of given perimeter, arXiv:1710.11245 [math.CO], 2017.

Cf. A008742 (triangles), A293818 (reflections allowed), A293821 (quadrilaterals), A293822 (pentagons), A293823 (hexagons).

Row sums of A293819 (k-gon triangle).

T[n_, k_] := DivisorSum[GCD[n, k], EulerPhi[#]*Binomial[n/#, k/#] &]/n - Binomial[Floor[n/2], k - 1];
a[n_] := Sum[T[n, k], {k, 3, n}]
Table[a[n], {n, 3, 40}] (* Jean-François Alcover, Jun 14 2018, after Andrew Howroyd *)

a(n) = sumdiv(n, d, eulerphi(n/d)*2^d)/n - 1 - 2^floor(n/2); \\ Andrew Howroyd, Nov 21 2017

a(n) = (Sum_{d|n} phi(n/d)*2^d)/n - 1 - 2^floor(n/2). - Andrew Howroyd, Nov 21 2017

A293819 Triangle read by rows of the number of integer-sided k-gons having perimeter n, modulo rotations but not reflections, for k=3..n.

Original entry on oeis.org

1, 0, 1, 1, 1, 1, 1, 2, 1, 1, 2, 4, 3, 1, 1, 1, 6, 6, 4, 1, 1, 4, 10, 13, 10, 4, 1, 1, 2, 12, 21, 21, 12, 5, 1, 1, 5, 20, 37, 41, 30, 15, 5, 1, 1, 4, 23, 51, 74, 65, 43, 19, 6, 1, 1, 7, 35, 84, 126, 131, 99, 55, 22, 6, 1, 1, 5, 38, 108, 196, 239, 216, 143, 73, 26, 7, 1, 1, 10, 56, 166, 314, 422, 428

Offset: 3

James East, Oct 16 2017

Rotations are counted only once, but reflections are considered different. For a k-gon to be nondegenerate, the longest side must be shorter than the sum of the remaining sides (equivalently, shorter than n/2). Column k=3 is A008742, column k=4 is A293821, column k=5 is A293822 and column k=6 is A293823.

A formula is given in Section 6 of the East and Niles article.

			For polygons having perimeter 7, there are: 2 triangles (331, 322), 4 quadrilaterals (3211, 3121, 3112, 2221), 3 pentagons (31111, 22111, 21211), 1 hexagon (211111) and 1 heptagon (1111111). Note that the quadrilaterals 3211 and 3112 are reflections of each other, but these are not rotationally equivalent.
The triangle begins:
n=3:  1;
n=4:  0,  1;
n=5:  1,  1,  1;
n=6:  1,  2,  1,  1;
n=7:  2,  4,  3,  1,  1;
n=8:  1,  6,  6,  4,  1,  1;
n=9:  4, 10, 13, 10,  4,  1,  1;
...

Andrew Howroyd, Rows n=3..52 of triangle, flattened
James East, Ron Niles, Integer polygons of given perimeter, arXiv:1710.11245 [math.CO], 2017.

Columns: A008742 (triangles), A293821 (quadrilaterals), A293822 (pentagons), A293823 (hexagons).
Row sums are A293820.
Same triangle with reflection allowed is A124287.

T[n_, k_] := DivisorSum[GCD[n, k], EulerPhi[#]*Binomial[n/#, k/#]&]/n - Binomial[Floor[n/2], k - 1];
Table[T[n, k], {n, 3, 16}, {k, 3, n}] // Flatten (* Jean-François Alcover, Jun 14 2018, translated from PARI *)

T(n,k)={sumdiv(gcd(n,k), d, eulerphi(d)*binomial(n/d,k/d))/n - binomial(floor(n/2), k-1)}
for(n=3, 10, for(k=3, n, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 21 2017

T(n,k) = (Sum_{d|gcd(n,k)} phi(d)*binomial(n/d, k/d))/n - binomial(floor(n/2), k-1). - Andrew Howroyd, Nov 21 2017

A293818 Number of integer-sided polygons having perimeter n, modulo rotations and reflections.

Original entry on oeis.org

1, 1, 3, 5, 10, 16, 32, 54, 102, 180, 336, 607, 1144, 2098, 3960, 7397, 14022, 26452, 50404, 95821, 183322, 350554, 673044, 1292634, 2489502, 4797694, 9264396, 17904220, 34652962, 67125898, 130182972, 252679320, 490918440, 954505718, 1857413460, 3616951513, 7048412792, 13744169104

Offset: 3

James East, Oct 16 2017

nonn

Rotations and reversals are counted only once. For a polygon to be nondegenerate, the longest side must be shorter than the sum of the remaining sides. These are row sums of A124287.
A formula is proved in Theorem 1.6 of the East and Niles article.
The same article shows that a(n) is asymptotic to 2^(n-1) / n.

			There are 10 polygons having perimeter 7: 2 triangles, 3 quadrilaterals, 3 pentagons, 1 hexagon and 1 heptagon.

Andrew Howroyd, Table of n, a(n) for n = 3..200
James East, Ron Niles, Integer polygons of given perimeter, arXiv:1710.11245 [math.CO], 2017.

Row sums of A124287 (k-gon triangle).

Cf. A293820 (polygons modulo rotations only).

a[n_] := DivisorSum[n, EulerPhi[n/#]*2^# &]/(2*n) + 2^Floor[(n - 3)/2] - If[Mod[n, 4] < 2, 3*2^Floor[(n - 4)/4], 2^Floor[(n + 2)/4] ];
Table[a[n], {n, 3, 40}] (* Jean-François Alcover, Jun 14 2018, after Andrew Howroyd *)

a(n)={sumdiv(n, d, eulerphi(n/d)*2^d)/(2*n) + 2^floor((n-3)/2) - if(n%4<2, 3*2^floor((n-4)/4), 2^floor((n+2)/4))} \\ Andrew Howroyd, Nov 21 2017

A289620 Triangle read by rows: T(n,r), 0 <= r <= n, is the number of idempotents of rank r in the twisted planar partition monoid PP_n^tau.

Original entry on oeis.org

1, 0, 1, 0, 5, 1, 0, 33, 10, 1, 0, 253, 93, 15, 1, 0, 2147, 880, 178, 20, 1, 0, 19593, 8599, 1982, 288, 25, 1, 0, 188837, 86762, 21723, 3684, 423, 30, 1, 0, 1899107, 900997, 238419, 44767, 6111, 583, 35, 1, 0, 19761209, 9595264, 2638114, 531656, 81606, 9388, 768, 40, 1, 0, 211447863, 104447385, 29503900, 6255952, 1044248, 136740, 13640, 978, 45, 1

Offset: 0

James East, Oct 05 2017

Values were computed using the Semigroups package for GAP.

Igor Dolinka, James East, Athanasios Evangelou, Desmond FitzGerald, Nicholas Ham, James Hyde, Nicholas Loughlin, Idempotent Statistics of the Motzkin and Jones Monoids, arXiv:1507.04838 [math.CO], 2015-2016.

Cf. A281441, A281442, A286867 (row sums).

A286867 Number of idempotents in the twisted planar partition monoid PP_n^tau.

Original entry on oeis.org

1, 1, 6, 44, 362, 3226, 30488, 301460, 3090020, 32618046, 345515557

Offset: 0

James East, Oct 05 2017

Values were computed using the Semigroups package for GAP.

Igor Dolinka, James East, Athanasios Evangelou, Desmond FitzGerald, Nicholas Ham, James Hyde, Nicholas Loughlin, Idempotent Statistics of the Motzkin and Jones Monoids, arXiv:1507.04838 [math.CO], 2015-2016.

Cf. A225797, A289620.

A281442 Triangle read by rows: T(n,r), 0 <= r <= n, is the number of idempotents of rank r in the Kauffman monoid K_n.

Original entry on oeis.org

1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 0, 0, 4, 0, 1, 0, 8, 0, 6, 0, 1, 0, 0, 22, 0, 8, 0, 1, 0, 42, 0, 40, 0, 10, 0, 1, 0, 0, 140, 0, 62, 0, 12, 0, 1, 0, 262, 0, 288, 0, 88, 0, 14, 0, 1, 0, 0, 992, 0, 492, 0, 118, 0, 16, 0, 1

Offset: 0

James East, Oct 05 2017

Values were computed using the Semigroups package for GAP.

T(n,r) is also the number of idempotent basis elements of rank r in the Temperley-Lieb algebra of degree n in the generic case (when the twisting parameter is not an m-th root of unity for any m <= n).

Igor Dolinka, James East, Athanasios Evangelou, Desmond FitzGerald, Nicholas Ham, James Hyde, Nicholas Loughlin, Idempotent Statistics of the Motzkin and Jones Monoids, arXiv:1507.04838 [math.CO], 2015-2016.

Cf. A281438 (row sums), A281441, A289620.

T(2n-1,1) = A005315(n). Empirical: T(2n,2) = A077056(n); T(n+2,n-2) = 2*A028875(n) for n>2. - Andrey Zabolotskiy, Oct 19 2017

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User: James East

A340914 Square array, read by rows. For n,d >= 0, a(n,d) is the number of congruences of the d-twisted partition monoid of degree n.

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A293822 Number of integer-sided pentagons having perimeter n, modulo rotations but not reflections.

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A293823 Number of integer-sided hexagons having perimeter n, modulo rotations but not reflections.

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A293821 Number of integer-sided quadrilaterals having perimeter n, modulo rotations but not reflections.

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A293820 Number of integer-sided polygons having perimeter n, modulo rotations but not reflections.

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A293819 Triangle read by rows of the number of integer-sided k-gons having perimeter n, modulo rotations but not reflections, for k=3..n.

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A293818 Number of integer-sided polygons having perimeter n, modulo rotations and reflections.

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