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
Examples
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 ... ...
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.
Programs
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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
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).
Comments