0,6

An up-jump j occurs at position i in p if p_{i} > p_{i-1} and j is the index of p_i in the increasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are larger than p_{i-1}. A down-jump j occurs at position i in p if p_{i} < p_{i-1} and j is the index of p_i in the decreasingly sorted list of those elements in {p_{i}, ..., p_{n}} that are smaller than p_{i-1}. First index in the lists is 1 here.

From David B. Wilson, Dec 14 2018: (Start)

T(n,k) equals the number of permutations p of [n] such that twice the sum of the leftward-down-jumps of p plus the number of descents of p equals k.

T(n,k) equals the number of cover-inclusive Dyck tilings whose lower boundary is the zig-zag path of order n (UD)^n, and which have k tiles.

A leftward-down-jump j occurs at position i in p if p_{i} > p_{i+1} and there are j positions k for which k<i and p_i > p_k > p_{i+1}.

Cover-inclusive Dyck tilings are defined in the Kenyon and Wilson link below. (End)

Alois P. Heinz, Rows n = 0..50, flattened

R. W. Kenyon, D. B. Wilson, Double-dimer pairings and skew Young diagrams, The Electronic Journal of Combinatorics 18(1) #P130, 2011.

J. S. Kim, K. Mészáros, G. Panova, and D. B. Wilson. Dyck tilings, increasing trees, descents, and inversions, Journal of Combinatorial Theory A 122:9-27, 2014.

Sum_{k>=0} k * T(n,k) = A005990(n).

T(4,0) = 1: 1234.

T(4,1) = 6: 1243, 1324, 1342, 2134, 2314, 2341.

T(4,2) = 6: 1432, 2143, 2431, 3214, 3241, 3421.

T(4,3) = 5: 1423, 2413, 3124, 3412, 4321.

T(4,4) = 4: 3142, 4213, 4231, 4312.

T(4,5) = 1: 4123.

T(4,6) = 1: 4132.

T(5,5) = 15: 15234, 25134, 31542, 35124, 41235, 42153, 42531, 43152, 45123, 53214, 53241, 53421, 54213, 54231, 54312.

Triangle T(n,k) begins:

1;

1;

1, 1;

1, 3, 1, 1;

1, 6, 6, 5, 4, 1, 1;

1, 10, 20, 20, 26, 15, 15, 6, 5, 1, 1;

1, 15, 50, 70, 105, 106, 104, 90, 65, 51, 27, 21, 7, 6, 1, 1;

b:= proc(u, o) option remember; expand(`if`(u+o=0, 1,

add(b(u-j, o+j-1)*x^(j-1), j=1..u)+

add(b(u+j-1, o-j)*x^(j-1), j=1..o)))

end:

T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(0, n)):

seq(T(n), n=0..10);

(* Generating function for tiles for Dyck tilings above the zigzag path of order n *)

(* Computed by looking at descents in the insertion sequence for the Dyck-tiling-ribbon bijection, described in the Kim-Meszaros-Panova-Wilson reference *)

(* Since it's above the zigzag, all insertion positions are even *)

(* When the second argument is specified, refines by position of last insertion *)

tilegen[n_, sn_] := tilegen[n, sn] = If[n == 0 || n == 1, 1,

Sum[tilegen[n - 1, j] If[j >= sn, t^(j - sn + 1), 1] //

Expand, {j, 0, 2 (n - 2), 2}]

];

tilegen[n_] := tilegen[n + 1, 2 n];

T[n_, k_] := Coefficient[tilegen[n], t, k]; (* David B. Wilson, Dec 14 2018 *)

Columns k=0-3 give: A000012, A000217(n-1) for n>0, A002415(n-1) for n>0, A291288(n-3) for n>0.

Row sums give A000142.

T(n,n) gives A289489.

Cf. A005990, A008292, A123125, A173018, A258829, A263776, A316292, A316293.

Sequence in context: A160752 A186024 A080002 * A297672 A256973 A058057

Adjacent sequences: A291719 A291720 A291721 * A291723 A291724 A291725

nonn,tabf

Alois P. Heinz, Aug 30 2017

approved