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%I #27 May 09 2019 05:04:57
%S 1,1,0,1,1,0,4,1,1,0,15,7,1,1,0,76,31,11,1,1,0,455,185,60,18,1,1,0,
%T 3186,1275,435,113,29,1,1,0,25487,10095,3473,1001,215,47,1,1,0,229384,
%U 90109,31315,9289,2299,406,76,1,1,0,2293839,895169,313227,95747,24610,5320,763,123,1,1,0
%N Number T(n,k) of permutations p of [n] such that n-k is the maximum of 0 and the number of elements in any integer interval [p(i)..i+n*[i<p(i)]]; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%C Mirror image of A324563.
%H Alois P. Heinz, <a href="/A324564/b324564.txt">Rows n = 0..23, flattened</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Interval_(mathematics)#Integer_intervals">Integer intervals</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Iverson_bracket">Iverson bracket</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permanent_(mathematics)">Permanent (mathematics)</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Symmetric_group">Symmetric group</a>
%e Triangle T(n,k) begins:
%e 1;
%e 1, 0;
%e 1, 1, 0;
%e 4, 1, 1, 0;
%e 15, 7, 1, 1, 0;
%e 76, 31, 11, 1, 1, 0;
%e 455, 185, 60, 18, 1, 1, 0;
%e 3186, 1275, 435, 113, 29, 1, 1, 0;
%e 25487, 10095, 3473, 1001, 215, 47, 1, 1, 0;
%e ...
%e Square array A(n,k) begins:
%e 1, 0, 0, 0, 0, 0, ...
%e 1, 1, 1, 1, 1, 1, ...
%e 1, 1, 1, 1, 1, 1, ...
%e 4, 7, 11, 18, 29, 47, ...
%e 15, 31, 60, 113, 215, 406, ...
%e 76, 185, 435, 1001, 2299, 5320, ...
%e 455, 1275, 3473, 9289, 24610, 65209, ...
%e 3186, 10095, 31315, 95747, 290203, 876865, ...
%e ...
%p b:= proc(n, k) option remember; `if`(k>n, 0, `if`(k=0, n!,
%p LinearAlgebra[Permanent](Matrix(n, (i, j)->
%p `if`(j>=i and k+j<n+i or i>k+j, 1, 0)))))
%p end:
%p # as triangle:
%p T:= (n, k)-> b(n, k)-b(n, k+1):
%p seq(seq(T(n, k), k=0..n), n=0..10);
%p # as array:
%p A:= (n, k)-> b(n+k, k)-b(n+k, k+1):
%p seq(seq(A(d-k, k), k=0..d), d=0..10);
%t b[n_, k_] := b[n, k] = If[k > n, 0, If[k == 0, n!, Permanent[Table[If[j >= i && k+j < n+i || i > k+j, 1, 0], {i, n}, {j, n}]]]];
%t (* as triangle: *)
%t T[n_, k_] := b[n, k] - b[n, k+1];
%t Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten
%t (* as array: *)
%t A[n_, k_] := b[n+k, k] - b[n+k, k+1];
%t Table[A[d-k, k], {d, 0, 10}, {k, 0, d}] // Flatten (* _Jean-François Alcover_, May 09 2019, after _Alois P. Heinz_ *)
%Y Columns k=0-10 give: A002467 (for n>0), A324621, A324622, A324623, A324624, A324625, A324626, A324627, A324628, A324629, A324630.
%Y Diagonals of the triangle (rows of the array) n=0, (1+2), 3-10 give: A000007, A000012, A000032 (for n>=3), A324631, A324632, A324633, A324634, A324635, A324636, A324637.
%Y Row sums give A000142.
%Y T(2n,n) or A(n,n) gives A324638.
%Y Cf. A002467, A324563.
%K nonn,tabl
%O 0,7
%A _Alois P. Heinz_, Mar 06 2019
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