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

0,1

Comments

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

Links

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.

Formula

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).

Example

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 ...
  ...

Prog

(PARI) 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

Crossrefs

Sequence in context: A209163 A124932 A248788 * A194232 A371567 A110042

Adjacent sequences: A340911 A340912 A340913 * A340915 A340916 A340917

Keyword

nonn,tabl

Author

James East, Mar 07 2021

Status

approved