OFFSET
0,5
COMMENTS
The triangle is the matrix inverse of the Riordan square (see A321620) generated by (1 + x - sqrt(1 - 6*x + x^2))/(4*x) (see A172094), where we take the absolute value of the terms.
T(n,k) is the number of evil-avoiding (2413, 3214, 4132, and 4213 avoiding) permutations of length (n+2) that start with 1 and whose inverse has k descents. - Donghyun Kim, Aug 16 2021
LINKS
Donghyun Kim and Lauren Williams, Schubert polynomials and the inhomogeneous TASEP on a ring, arXiv:2102.00560 [math.CO], 2021.
EXAMPLE
Triangle starts:
[0] [1]
[1] [1, 1]
[2] [1, 4, 1]
[3] [1, 11, 7, 1]
[4] [1, 26, 30, 10, 1]
[5] [1, 57, 102, 58, 13, 1]
[6] [1, 120, 303, 256, 95, 16, 1]
[7] [1, 247, 825, 955, 515, 141, 19, 1]
[8] [1, 502, 2116, 3178, 2310, 906, 196, 22, 1]
[9] [1, 1013, 5200, 9740, 9078, 4746, 1456, 260, 25, 1]
...
Seen as a square array (the triangle is formed by descending antidiagonals):
1, 1, 1, 1, 1, 1, 1, 1, 1, ... [A000012]
1, 4, 11, 26, 57, 120, 247, 502, 1013, ... [A000295]
1, 7, 30, 102, 303, 825, 2116, 5200, 12381, ... [A045889]
1, 10, 58, 256, 955, 3178, 9740, 28064, 77093, ... [A055583]
1, 13, 95, 515, 2310, 9078, 32354, 106970, 333295, ...
1, 16, 141, 906, 4746, 21504, 87374, 326084, 1136799, ...
1, 19, 196, 1456, 8722, 44758, 204204, 849180, 3275931, ...
MAPLE
gf := k -> 1/(((1-2*x)^k)*(1-x)^(k+1)): ser := k -> series(gf(k), x, 32):
# Prints the triangle:
seq(lprint(seq(coeff(ser(k), x, n-k), k=0..n)), n=0..6);
# Prints the square array:
seq(lprint(seq(coeff(ser(k), x, n), n=0..8)), k=0..6);
MATHEMATICA
(* The function RiordanSquare is defined in A321620; returns the triangle as a lower triangular matrix. *)
M := RiordanSquare[(1 + x - Sqrt[1 - 6 x + x^2])/(4 x), 9];
Abs[#] & /@ Inverse[PadRight[M]]
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Peter Luschny, Feb 03 2020
STATUS
approved