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A322481 Permutation breadth triangle: B(n,k) is the number of permutations w in S_n with breadth(w) = k, where breadth(w) = min({ |i-j|+|w(i)-w(j)| : 1 <= i < j <= n }).

%I A322481 #20 Dec 12 2018 11:09:47
%S A322481 0,0,2,0,6,0,0,22,2,0,0,106,14,0,0,0,630,90,0,0,0,0,4394,644,2,0,0,0,
%T A322481 0,35078,5222,20,0,0,0,0,0,315258,47464,158,0,0,0,0,0,0,3149494,
%U A322481 477346,1960,0,0,0,0,0,0
%N A322481 Permutation breadth triangle: B(n,k) is the number of permutations w in S_n with breadth(w) = k, where breadth(w) = min({ |i-j|+|w(i)-w(j)| : 1 <= i < j <= n }).
%C A322481 B(n,1) = 0 for all n, because for any 1<=i,j<=n and any w in S_n, 2 <= |i-j|+|w(i)-w(j)| <= breadth(w).
%H A322481 D. Bevan, C. Homberger, and B. E. Tenner, <a href="https://arxiv.org/abs/1608.06931">Prolific permutations and permuted packings: downsets containing many large patterns</a>, arXiv:1608.06931 [math.CO], 2016_2017; J. Combin. Theory A., 153:98-121, 2018.
%e A322481 For n=4, k=3, the B(4,3) = 2 permutations in S_4 with breadth 3 are [2,4,1,3] and [3,1,4,2] in one-line notation.
%e A322481 Triangle: B(n,k) begins:
%e A322481   0;
%e A322481   0,       2;
%e A322481   0,       6,      0;
%e A322481   0,      22,      2,    0;
%e A322481   0,     106,     14,    0, 0;
%e A322481   0,     630,     90,    0, 0, 0;
%e A322481   0,    4394,    644,    2, 0, 0, 0;
%e A322481   0,   35078,   5222,   20, 0, 0, 0, 0;
%e A322481   0,  315258,  47464,  158, 0, 0, 0, 0, 0;
%e A322481   0, 3149494, 477346, 1960, 0, 0, 0, 0, 0, 0;
%Y A322481 Column k=2 gives A129535.
%Y A322481 Row sums give A000142 (for n>1).
%K A322481 nonn,tabl
%O A322481 1,3
%A A322481 _Jordan Weaver_, Dec 10 2018