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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. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

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

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)

EXAMPLE

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, ...

  ...

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

CROSSREFS

Row sums give A027686.

Right border gives A003121(n-1).

Sequence in context: A102888 A071558 A179621 * A136709 A254969 A137239

Adjacent sequences:  A282695 A282696 A282697 * A282699 A282700 A282701

KEYWORD

nonn,tabf

AUTHOR

N. J. A. Sloane, Feb 25 2017

EXTENSIONS

More terms from Alois P. Heinz, Jan 02 2018

STATUS

approved

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Last modified June 20 10:53 EDT 2021. Contains 345162 sequences. (Running on oeis4.)