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 A324225 Total number T(n,k) of 1's in falling diagonals with index k in all n X n permutation matrices; triangle T(n,k), n>=1, 1-n<=k<=n-1, read by rows. 3
 1, 1, 2, 1, 2, 4, 6, 4, 2, 6, 12, 18, 24, 18, 12, 6, 24, 48, 72, 96, 120, 96, 72, 48, 24, 120, 240, 360, 480, 600, 720, 600, 480, 360, 240, 120, 720, 1440, 2160, 2880, 3600, 4320, 5040, 4320, 3600, 2880, 2160, 1440, 720, 5040, 10080, 15120, 20160, 25200, 30240, 35280, 40320, 35280, 30240, 25200, 20160, 15120, 10080, 5040 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS T(n,k) is the number of occurrences of k in all (signed) displacement lists [p(i)-i, i=1..n] of permutations p of [n]. LINKS Alois P. Heinz, Rows n = 1..100, flattened Wikipedia, Permutation Wikipedia, Permutation matrix FORMULA T(n,k) = T(n,-k). T(n,k) = (n-t)*(n-1)! if t < n with t = |k|, T(n,k) = 0 otherwise. T(n,k) = |k|! * A324224(n,k). E.g.f. of column k: x^t/t * hypergeom([2, t], [t+1], x) with t = |k|+1. |T(n,k)-T(n,k-1)| = (n-1)! for k = 1-n..n. EXAMPLE The 6 permutations p of [3]: 123, 132, 213, 231, 312, 321 have (signed) displacement lists [p(i)-i, i=1..3]: [0,0,0], [0,1,-1], [1,-1,0], [1,1,-2], [2,-1,-1], [2,0,-2], representing the indices of falling diagonals of 1's in the permutation matrices   [1    ]  [1    ]  [  1  ]  [  1  ]  [    1]  [    1]   [  1  ]  [    1]  [1    ]  [    1]  [1    ]  [  1  ]   [    1]  [  1  ]  [    1]  [1    ]  [  1  ]  [1    ] , respectively. Indices -2 and 2 occur twice, -1 and 1 occur four times, and 0 occurs six times. So row n=3 is [2, 4, 6, 4, 2]. Triangle T(n,k) begins:   :                             1                           ;   :                        1,   2,   1                      ;   :                   2,   4,   6,   4,   2                 ;   :              6,  12,  18,  24,  18,  12,   6            ;   :        24,  48,  72,  96, 120,  96,  72,  48,  24       ;   :  120, 240, 360, 480, 600, 720, 600, 480, 360, 240, 120  ; MAPLE b:= proc(s, c) option remember; (n-> `if`(n=0, c,       add(b(s minus {i}, c+x^(n-i)), i=s)))(nops(s))     end: T:= n-> (p-> seq(coeff(p, x, i), i=1-n..n-1))(b({\$1..n}, 0)): seq(T(n), n=1..8); # second Maple program: egf:= k-> (t-> x^t/t*hypergeom([2, t], [t+1], x))(abs(k)+1): T:= (n, k)-> n! * coeff(series(egf(k), x, n+1), x, n): seq(seq(T(n, k), k=1-n..n-1), n=1..8); # third Maple program: T:= (n, k)-> (t-> `if`(t1. Cf. A152883 (right half of this triangle without center column), A162608 (left half of this triangle), A306461, A324224. Sequence in context: A139145 A201635 A179787 * A214739 A296159 A283334 Adjacent sequences:  A324222 A324223 A324224 * A324226 A324227 A324228 KEYWORD nonn,look,tabf AUTHOR Alois P. Heinz, Feb 18 2019 STATUS approved

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Last modified June 24 02:27 EDT 2021. Contains 345413 sequences. (Running on oeis4.)