A094718 Array T read by antidiagonals: T(n,k) is the number of involutions avoiding 132 and 12...k.

0, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 2, 2, 1, 0, 1, 2, 3, 4, 1, 0, 1, 2, 3, 5, 4, 1, 0, 1, 2, 3, 6, 8, 8, 1, 0, 1, 2, 3, 6, 9, 13, 8, 1, 0, 1, 2, 3, 6, 10, 18, 21, 16, 1, 0, 1, 2, 3, 6, 10, 19, 27, 34, 16, 1, 0, 1, 2, 3, 6, 10, 20, 33, 54, 55, 32, 1, 0, 1, 2, 3, 6, 10, 20, 34, 61, 81, 89, 32, 1

Offset: 1

Author

Ralf Stephan, May 23 2004

Keywords

• nonn
• tabl

Comments

Also, number of paths along a corridor with width k, starting from one side (from H. Bottomley's comment in A061551).

Rows converge to binomial(n,floor(n/2)) (A001405).

Note that the rows and columns start at 1, which for example obscures the fact that the first row refers to A000007 and not to A000004. A better choice is the indexing 0 <= k and 0 <= n. The Maple program below uses this indexing and builds only on the roots of -1. - Peter Luschny, Sep 17 2020

Examples

			Array begins
  0   0   0   0   0   0   0   0   0   0 ...
  1   1   1   1   1   1   1   1   1   1 ...
  1   2   2   4   4   8   8  16  16  32 ...
  1   2   3   5   8  13  21  34  55  89 ...
  1   2   3   6   9  18  27  54  81 162 ...
  1   2   3   6  10  19  33  61 108 197 ...
  1   2   3   6  10  20  34  68 116 232 ...
  1   2   3   6  10  20  35  69 124 241 ...
  1   2   3   6  10  20  35  70 125 250 ...
  1   2   3   6  10  20  35  70 126 251 ...
  ...
		

Links

• Shaun V. Ault and Charles Kicey, Counting paths in corridors using circular Pascal arrays, Discrete Mathematics, Volume 332, October 2014, Pages 45-54.
• Nachum Dershowitz, Between Broadway and the Hudson, arXiv:2006.06516 [math.CO], 2020.
• Robert Dutton Fray and David Paul Roselle, Weighted lattice paths, Pacific Journal of Mathematics, 37(1) (1971), 85-96.
• O. Guibert and T. Mansour, Restricted 132-involutions, Séminaire Lotharingien de Combinatoire, B48a (2002), 23 pp.
• T. Mansour, Restricted even permutations and Chebyshev polynomials, arXiv:math/0302014 [math.CO], 2003.

Crossrefs

Rows 3-8 are A016116, A000045, A038754, A028495, A030436, A061551.

Main diagonal is A014495, antidiagonal sums are in A094719.

Cf. A080934 (permutations).

Programs

Maple

X := (j, n) -> (-1)^(j/(n+1)) - (-1)^((n-j+1)/(n+1)):
R := n -> select(k -> type(k, odd), [$(1..n)]):
T := (n, k) -> add((2 + X(j,n))*X(j,n)^k, j in R(n))/(n+1):
seq(lprint([n], seq(simplify(T(n, k)), k=0..10)), n=0..9); # Peter Luschny, Sep 17 2020

Mathematica

U = ChebyshevU;
m = maxExponent = 14;
row[1] = Array[0&, m];
row[k_] := 1/(x U[k, 1/(2x)])*Sum[U[j, 1/(2x)], {j, 0, k-1}] + O[x]^m // CoefficientList[#, x]& // Rest;
T = Table[row[n], {n, 1, m}];
Table[T[[n-k+1, k]], {n, 1, m-1}, {k, 1, n}] // Flatten (* Jean-François Alcover, Nov 17 2018 *)

Formula

G.f. for k-th row: 1/(x*U(k, 1/(2*x))) * Sum_{j=0..k-1} U(j, 1/(2*x)), with U(k, x) the Chebyshev polynomials of second kind. [Clarified by Jean-François Alcover, Nov 17 2018]

T(n, k) = (1/(n+1))*Sum_{j=1..n, j odd} (2 + [j, n]) * [j, n]^k where [j, n] := (-1)^(j/(n+1)) - (-1)^((n-j+1)/(n+1)). - Peter Luschny, Sep 17 2020