A282698 - OEIS

%I #34 Jan 22 2018 10:19:20

%S 1,1,1,1,1,4,2,2,1,10,22,22,18,13,12,1,20,112,232,382,348,456,390,420,

%T 334,286,1,35,392,1744,4474,8435,12732,17337,21158,27853,33940,41230,

%U 45048,50752,41826,33592,1,56,1092,9220,40414,123704,276324,550932,917884

%N Irregular triangle read by rows: row n gives numbers of maximal chains of lengths n-1, n, n+1, ... in the Tamari lattice T_n.

%C Nelson (2017) gives first nine columns of the transposed triangle.

%H Alois P. Heinz, <a href="/A282698/b282698.txt">Rows n = 1..14, flattened</a>

%H Luke Nelson, <a href="https://doi.org/10.1016/j.disc.2016.11.030">A recursion on maximal chains in the Tamari lattices</a>, Discrete Mathematics 340.4 (2017): 661-677.

%H Luke Nelson, <a href="https://arxiv.org/abs/1709.02987">A recursion on maximal chains in the Tamari lattices</a>, arXiv:1709.02987 [math.CO], (2017)

%e Triangle begins:

%e   1;

%e   1;

%e   1,  1;

%e   1,  4,   2,   2;

%e   1, 10,  22,  22,  18,  13,  12;

%e   1, 20, 112, 232, 382, 348, 456, 390, 420, 334, 286;

%e   ...

%e The transposed triangle, as given by Nelson, begins:

%e   1, 1, 1, 1,  1,   1,     1,        1,          1, ...

%e         1, 4, 10,  20,    35,       56,         84, ...

%e            2, 22, 112,   392,     1092,       2604, ...

%e            2, 22, 232,  1744,     9220,      37444, ...

%e               18, 382,  4474,    40414,     280214, ...

%e               13, 348,  8435,   123704,    1321879, ...

%e               12, 456, 12732,   276324,    4578596, ...

%e                   390, 17337,   550932,   12512827, ...

%e                   420, 21158,   917884,   29499764, ...

%e                   334, 27853,  1510834,   62132126, ...

%e                   286, 33940,  2166460,  120837274, ...

%e                        41230,  3370312,  221484557, ...

%e                        45048,  4810150,  393364848, ...

%e                        50752,  7264302,  666955139, ...

%e                        41826, 10435954, 1134705692, ...

%e                        33592, 15227802, 1933708535, ...

%e   ...

%p s:= proc(n) s(n):=`if`(n=0, [], [s(n-1), []]) end:

%p f:= l-> l=[] or l[1]=[] and f(l[2]):

%p v:= proc(l) v(l):=`if`(f(l), [], [`if`(l[1]<>[],

%p       [l[1][1], [l[1][2], l[2]]], [][]),

%p       seq([w, l[2]], w=v(l[1])), seq([l[1], w], w=v(l[2]))])

%p     end:

%p p:= proc(l) p(l):=`if`(f(l), 1, add(expand(x*p(w)), w=v(l))) end:

%p T:= n-> (h-> seq(coeff(h, x, i), i=ldegree(h)..degree(h)))(p(s(n))):

%p seq(T(n), n=1..8);  # _Alois P. Heinz_, Jan 02 2018

%Y Row sums give A027686.

%Y Right border gives A003121(n-1).

%K nonn,tabf

%O 1,6

%A _N. J. A. Sloane_, Feb 25 2017

%E More terms from _Alois P. Heinz_, Jan 02 2018