A231601 -id:A231601 - cp's OEIS frontend

A285672 Number of permutations p of [n] avoiding consecutive odd sums i+p(i), (i+1)+p(i+1) for all i in [n-1].

1, 1, 1, 2, 8, 36, 180, 1008, 6336, 46080, 374400, 3369600, 32659200, 344736000, 3886444800, 47348582400, 611264102400, 8442272563200, 122595843686400, 1890952003584000, 30510694932480000, 520011800985600000, 9231875243458560000, 172292221923655680000

Offset: 0

Alois P. Heinz, Apr 24 2017

nonn

			a(0) = 1: the empty permutation.
a(1) = 1: 1.
a(2) = 1: 12.
a(3) = 2: 123, 321.
a(4) = 8: 1234, 1432, 2413, 2431, 3214, 3412, 4213, 4231.
a(5) = 36: 12345, 12543, 13524, 13542, 14325, 14523, 15324, 15342, 24135, 24153, 24315, 24351, 24513, 24531, 31524, 31542, 32145, 32541, 34125, 34521, 35124, 35142, 42135, 42153, 42315, 42351, 42513, 42531, 51324, 51342, 52143, 52341, 53124, 53142, 54123, 54321.

Alois P. Heinz, Table of n, a(n) for n = 0..466

Cf. A199660, A231601.

b:= proc(n, i, j, p, t) option remember; `if`(n=0, 1,
     `if`(i=0 or t=1 and p=1, 0, i*b(n-1, i-1, j, 1-p, p))+
     `if`(j=0 or t=1 and p=0, 0, j*b(n-1, i, j-1, 1-p, 1-p)))
    end:
a:= n-> b(n, floor(n/2), ceil(n/2), 1, 0):
seq(a(n), n=0..25);

b[n_, i_, j_, p_, t_] := b[n, i, j, p, t] =
     If[n==0, 1, If[i==0 || t ==1 && p==1, 0, i*b[n-1, i-1, j, 1-p, p]] +
     If[j==0 || t==1 && p==0, 0, j*b[n-1, i, j-1, 1-p, 1-p]]];
a[n_] := b[n, Floor[n/2], Ceiling[n/2], 1, 0];
Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Aug 30 2021, after Alois P. Heinz *)

A231777 Number T(n,k) of permutations of [n] with exactly k ascents from odd to even numbers; triangle T(n,k), n>=0, 0<=k<=floor(n/2), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 4, 2, 8, 14, 2, 54, 60, 6, 162, 402, 150, 6, 1536, 2712, 768, 24, 6144, 19704, 12744, 1704, 24, 75000, 183120, 94320, 10320, 120, 375000, 1473720, 1392720, 365520, 21720, 120, 5598720, 17522640, 13631040, 3011040, 152640, 720, 33592320, 156250800

Offset: 0

Alois P. Heinz, Nov 13 2013

			T(4,0) = 8: 1324, 2413, 2431, 3241, 4132, 4213, 4231, 4321.
T(4,1) = 14: 1243, 1342, 1423, 1432, 2134, 2143, 2314, 2341, 3124, 3142, 3214, 3421, 4123, 4312.
T(4,2) = 2: 1234, 3412.
T(5,2) = 6: 12345, 12534, 34125, 34512, 51234, 53412.
T(6,3) = 6: 123456, 125634, 341256, 345612, 561234, 563412.
Triangle T(n,k) begins:
:  0 :      1;
:  1 :      1;
:  2 :      1,       1;
:  3 :      4,       2;
:  4 :      8,      14,       2;
:  5 :     54,      60,       6;
:  6 :    162,     402,     150,      6;
:  7 :   1536,    2712,     768,     24;
:  8 :   6144,   19704,   12744,   1704,    24;
:  9 :  75000,  183120,   94320,  10320,   120;
: 10 : 375000, 1473720, 1392720, 365520, 21720, 120;

Alois P. Heinz, Rows n = 0..23, flattened

Column k=0 gives: A231601.
Row sums and T(2n,n) give: A000142.
T(n,floor(n/2)) gives: A081123(n+1).
Cf. A004526.

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A285672 Number of permutations p of [n] avoiding consecutive odd sums i+p(i), (i+1)+p(i+1) for all i in [n-1].

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