%I #38 Jan 06 2024 14:32:14

%S 2,3,3,9,7,4,12,43,14,5,16,76,136,24,6,19,134,329,334,37,7,22,188,773,

%T 1105,696,53,8,25,251,1281,3456,3100,1294,72,9,28,323,1969,6754,12806,

%U 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

%N 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.

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

%H Andrew Howroyd, <a href="/A340914/b340914.txt">Table of n, a(n) for n = 0..1325</a> (first 51 antidiagonals)

%H James East and Nik Ruškuc, <a href="https://arxiv.org/abs/2010.09288">Properties of congruences of twisted partition monoids and their lattices</a>, arXiv:2010.09288 [math.RA], 2020-2021.

%F a(0,d) = d+2,

%F a(1,d) = (3*d^2+5*d+6)/2,

%F a(2,d) = (13*d^4+106*d^3+299*d^2+398*d+216)/24,

%F 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,

%F 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.

%F For fixed d >= 0, a(n,d) is asymptotic to (3*n)^(d+1) / (d+1)!.

%F For fixed n >= 4, a(n,d) is asymptotic to 13*d^(3*n-1) / (3*n-1)!.

%F 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).

%e Array begins:

%e =========================================================

%e n\d | 0 1 2 3 4 5 6 7 ...

%e ----+----------------------------------------------------

%e 0 | 2 3 4 5 6 7 8 9 ...

%e 1 | 3 7 14 24 37 53 72 94 ...

%e 2 | 9 43 136 334 696 1294 2213 3551 ...

%e 3 | 12 76 329 1105 3100 7608 16842 34353 ...

%e 4 | 16 134 773 3456 12806 41054 117273 304889 ...

%e 5 | 19 188 1281 6754 29413 110312 366724 1103538 ...

%e 6 | 22 251 1969 11930 59547 255132 965409 3293916 ...

%e 7 | 25 323 2864 19578 110012 529298 2242845 8544569 ...

%e ...

%o (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

%K nonn,tabl

%O 0,1

%A _James East_, Mar 07 2021