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 A262372 Number T(n,k) of ordered pairs (p,q) of permutations of [n] with equal up-down signatures and p(1)=q(1)=k if n>0; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 13
 1, 0, 1, 0, 1, 1, 0, 2, 2, 2, 0, 10, 8, 8, 10, 0, 88, 68, 64, 68, 88, 0, 1216, 952, 852, 852, 952, 1216, 0, 24176, 19312, 17008, 16328, 17008, 19312, 24176, 0, 654424, 533544, 467696, 438496, 438496, 467696, 533544, 654424 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 LINKS Alois P. Heinz, Rows n = 0..100, flattened EXAMPLE T(4,1) = 10: (1234,1234), (1243,1243), (1243,1342), (1324,1324), (1324,1423), (1342,1243), (1342,1342), (1423,1324), (1423,1423), (1432,1432). T(4,2) = 8: (2134,2134), (2143,2143), (2314,2314), (2314,2413), (2341,2341), (2413,2314), (2413,2413), (2431,2431). T(4,3) = 8: (3124,3124), (3142,3142), (3142,3241), (3214,3214), (3241,3142), (3241,3241), (3412,3412), (3421,3421). T(4,4) = 10: (4123,4123), (4132,4132), (4132,4231), (4213,4213), (4213,4312), (4231,4132), (4231,4231), (4312,4213), (4312,4312), (4321,4321). Triangle T(n,k) begins:   1   0,     1;   0,     1,     1;   0,     2,     2,     2;   0,    10,     8,     8,    10;   0,    88,    68,    64,    68,    88;   0,  1216,   952,   852,   852,   952,  1216;   0, 24176, 19312, 17008, 16328, 17008, 19312, 24176; MAPLE b:= proc(u, o, h) option remember; `if`(u+o=0, 1,       add(add(b(u-j, o+j-1, h+i-1), i=1..u+o-h), j=1..u)+       add(add(b(u+j-1, o-j, h-i), i=1..h), j=1..o))     end: T:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), b(k-1, n-k, n-k)): seq(seq(T(n, k), k=0..n), n=0..10); MATHEMATICA b[u_, o_, h_] := b[u, o, h] = If[u + o == 0, 1,   Sum[b[u - j, o + j - 1, h + i - 1], {i, 1, u + o - h}, {j, 1, u}] +   Sum[b[u + j - 1, o - j, h - i], {i, 1, h}, {j, 1, o}]]; T[n_, k_] := If[k == 0, If[n == 0, 1, 0], b[k - 1, n - k, n - k]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 05 2019, after Alois P. Heinz *) CROSSREFS Main diaginal and column k=1 give A060350(n-1) for n>0. Columns k=0,2-10 give: A000007, A262479, A321059, A321060, A321061, A321062, A321063, A321064, A321065, A321066. Row sums give A262234. T(2n,n) gives A262379. Sequence in context: A307520 A265648 A181230 * A292520 A131079 A334889 Adjacent sequences:  A262369 A262370 A262371 * A262373 A262374 A262375 KEYWORD nonn,look,tabl AUTHOR Alois P. Heinz, Sep 20 2015 STATUS approved

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Last modified May 23 12:48 EDT 2022. Contains 353975 sequences. (Running on oeis4.)