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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

%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

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Last modified March 28 18:04 EDT 2024. Contains 371254 sequences. (Running on oeis4.)