A098157 - OEIS

%I #28 Aug 29 2024 23:33:51

%S 1,1,1,0,3,1,0,1,6,1,0,0,5,10,1,0,0,1,15,15,1,0,0,0,7,35,21,1,0,0,0,1,

%T 28,70,28,1,0,0,0,0,9,84,126,36,1,0,0,0,0,1,45,210,210,45,1,0,0,0,0,0,

%U 11,165,462,330,55,1,0,0,0,0,0,1,66,495,924,495,66,1,0,0,0,0,0,0,13,286,1287,1716,715,78,1

%N Triangle T(n,k) with diagonals T(n,n-k) = binomial(n+1,2k).

%C Row sums are A000079. Diagonal sums are A062200. Inverse is A065547, less the first column.

%C Number of permutations of length n avoiding simultaneously the patterns 123 and 132 with k descents. A descent in a permutation a(1)a(2)...a(n) is position i such that a(i)>a(i+1). - _Tian Han_, Nov 16 2023

%H T. Han and S. Kitaev, <a href="https://arxiv.org/abs/2311.02974">Joint distributions of statistics over permutations avoiding two patterns of length 3</a>, arXiv:2311.02974 [math.CO], 2023.

%F T(n, k) = binomial(n+1, 2(n-k)) with 0 <= k <= n.

%F G.f.: (1 + x - q*x)/(1 - 2*q*x - q*x^2 + q^2*x^2). - _Tian Han_, Nov 16 2023

%e Rows begin:

%e  {1},

%e  {1,1},

%e  {0,3,1},

%e  {0,1,6,1},

%e  {0,0,5,10,1},

%e  {0,0,1,15,15,1},

%e   ...

%t Table[Binomial[n+1, 2(n-k)],{n,0,11},{k,0,n}]//Flatten (* _Stefano Spezia_, Nov 16 2023 *)

%Y Cf. A000079, A062200, A065547.

%K easy,nonn,tabl

%O 0,5

%A _Paul Barry_, Aug 29 2004