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
KEYWORD
nonn,tabl
AUTHOR
James East, Mar 07 2021
STATUS
approved