A282698 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.

1, 1, 1, 1, 1, 4, 2, 2, 1, 10, 22, 22, 18, 13, 12, 1, 20, 112, 232, 382, 348, 456, 390, 420, 334, 286, 1, 35, 392, 1744, 4474, 8435, 12732, 17337, 21158, 27853, 33940, 41230, 45048, 50752, 41826, 33592, 1, 56, 1092, 9220, 40414, 123704, 276324, 550932, 917884

Offset: 1

Author

N. J. A. Sloane, Feb 25 2017

Keywords

• nonn
• tabf

Comments

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

Examples

			Triangle begins:
  1;
  1;
  1,  1;
  1,  4,   2,   2;
  1, 10,  22,  22,  18,  13,  12;
  1, 20, 112, 232, 382, 348, 456, 390, 420, 334, 286;
  ...
The transposed triangle, as given by Nelson, begins:
  1, 1, 1, 1,  1,   1,     1,        1,          1, ...
        1, 4, 10,  20,    35,       56,         84, ...
           2, 22, 112,   392,     1092,       2604, ...
           2, 22, 232,  1744,     9220,      37444, ...
              18, 382,  4474,    40414,     280214, ...
              13, 348,  8435,   123704,    1321879, ...
              12, 456, 12732,   276324,    4578596, ...
                  390, 17337,   550932,   12512827, ...
                  420, 21158,   917884,   29499764, ...
                  334, 27853,  1510834,   62132126, ...
                  286, 33940,  2166460,  120837274, ...
                       41230,  3370312,  221484557, ...
                       45048,  4810150,  393364848, ...
                       50752,  7264302,  666955139, ...
                       41826, 10435954, 1134705692, ...
                       33592, 15227802, 1933708535, ...
  ...
		

Links

• Alois P. Heinz, Rows n = 1..14, flattened
• Luke Nelson, A recursion on maximal chains in the Tamari lattices, Discrete Mathematics 340.4 (2017): 661-677.
• Luke Nelson, A recursion on maximal chains in the Tamari lattices, arXiv:1709.02987 [math.CO], (2017)

Crossrefs

Row sums give A027686.

Right border gives A003121(n-1).

Programs

Maple

s:= proc(n) s(n):=`if`(n=0, [], [s(n-1), []]) end:
f:= l-> l=[] or l[1]=[] and f(l[2]):
v:= proc(l) v(l):=`if`(f(l), [], [`if`(l[1]<>[],
      [l[1][1], [l[1][2], l[2]]], [][]),
      seq([w, l[2]], w=v(l[1])), seq([l[1], w], w=v(l[2]))])
    end:
p:= proc(l) p(l):=`if`(f(l), 1, add(expand(x*p(w)), w=v(l))) end:
T:= n-> (h-> seq(coeff(h, x, i), i=ldegree(h)..degree(h)))(p(s(n))):
seq(T(n), n=1..8);  # Alois P. Heinz, Jan 02 2018

Extensions

More terms from Alois P. Heinz, Jan 02 2018