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A350158 The distribution of the distance from the first weak subcedance to 1 on permutations.

%I A350158 #16 Dec 17 2021 20:55:08
%S A350158 1,2,0,5,1,0,17,5,2,0,75,23,16,6,0,407,119,104,66,24,0,2619,719,688,
%T A350158 558,336,120,0,19487,5039,4976,4554,3504,2040,720,0,164571,40319,
%U A350158 40192,38862,34176,25320,14400,5040,0,1555007,362879,362624,358506,338304,287880,207360,115920,40320,0
%N A350158 The distribution of the distance from the first weak subcedance to 1 on permutations.
%C A350158 Triangular array read by rows. For 0 <= k <= n-1, T(n,k) is the number of permutations of [n] for which the difference between the position of 1 and the position of the first weak subcedance is k. A weak subcedance of a permutation pi is an entry pi(i) such that pi(i) <= i. See link.
%H A350158 David Callan, <a href="/A350158/a350158.pdf">The distribution of the distance from the first weak subcedance to 1 on permutations</a>
%F A350158 T(1,0) = 1, T(n,0) = 2*(n-1)! + Sum_{j=1..n-2} j^(n-j-1)*j! for n >= 2, T(n,k) = (n-1)! - k^(n-k-1)*k! for 1 <= k <= n-1.
%e A350158 Triangle T(n,k) begins:
%e A350158        1;
%e A350158        2,     0;
%e A350158        5,     1,     0;
%e A350158       17,     5,     2,     0;
%e A350158       75,    23,    16,     6,     0;
%e A350158      407,   119,   104,    66,    24,     0;
%e A350158     2619,   719,   688,   558,   336,   120,     0;
%e A350158    19487,  5039,  4976,  4554,  3504,  2040,   720,    0;
%e A350158   164571, 40319, 40192, 38862, 34176, 25320, 14400, 5040, 0;
%e A350158   ...
%t A350158 a[1, 0] = 1;
%t A350158 a[n_, 0] /; n >= 2 := 2 (n - 1)! + Sum[k^(n - k - 1) k!, {k, 1, n - 2}];
%t A350158 a[n_, k_] /; n > k >= 1 := (n - 1)! - k^(n - k - 1) k!;
%t A350158 Flatten[Table[a[n, k], {n, 10}, {k, 0, n - 1}]]
%Y A350158 Cf. A129591 is the first column.
%Y A350158 Row sums give A000142.
%K A350158 nonn,tabl
%O A350158 1,2
%A A350158 _David Callan_, Dec 17 2021