A185356 Triangle read by rows: number of type B alternating permutations according to their last value.

0, 1, 0, 1, 0, 1, 0, 1, 2, 4, 4, 3, 0, 3, 2, 0, 0, 4, 8, 11, 0, 11, 14, 16, 16, 80, 80, 76, 68, 57, 0, 57, 46, 32, 16, 0, 0, 80, 160, 236, 304, 361, 0, 361, 418, 464, 496, 512, 512, 3904, 3904, 3824, 3664, 3428, 3124, 2763, 0, 2763, 2402, 1984, 1520, 1024, 512, 0

Offset

0,9

Comments

"The table counting type B alternating permutations by their last value is obtained by the following algorithm: first separate the picture by the column p = 0 and then compute two triangles. Put 1 at the top of each triangle and compute the rest as follows: fill the second row of the left (resp. right) triangle as the sum of the elements of the first row (resp. strictly) to their left. Then fill the third row of the right (resp. left) triangle as the sum of the elements of the previous row (resp. strictly) to their right. Compute all rows successively by reading from left to right and right to left alternately." [Joshua-Verges et al.]

Links

Table of n, a(n) for n=0..63.

M. Josuat-Vergès, J.-C. Novelli and J.-Y. Thibon, The algebraic combinatorics of snakes, arXiv preprint arXiv:1110.5272 [math.CO], 2011.

Example

Triangle begins:
                       0
                     1 0 1
                 0   1 0 1   2
             4   4   3 0 3   2   0
         0   4   8  11 0 11  14  16  16
    80  80  76  68  57 0 57  46  32  16  0
  0 80 160 236 304 361 0 361 418 464 496 512 512

Prog

(PARI) T(n, k) = {if ((k==0), return(0)); if (n==1, if (abs(k)==1, return(1))); if (n%2, if (k<0, sum(j=k+1, n-1, T(n-1, j)), sum(j=k, n-1, T(n-1, j))), if (k<0, sum(j=-n+1, k, T(n-1, j)), sum(j=-n+1, k-1, T(n-1, j)))); }
tabf(nn) = {for (n=0, nn, for (k=-n, n, print1(T(n, k), ", "); ); print; ); } \\ Michel Marcus, Jun 03 2020

Crossrefs

See A202690 for another version.

See A010094 and A008281 for type A permutations.

Cf. A000831 (row sums, for n>0).

Cf. A001586 for the middle coefficients.

Sequence in context: A188424 A220588 A346262 * A008777 A306691 A354954

Adjacent sequences: A185353 A185354 A185355 * A185357 A185358 A185359

Keyword

nonn,tabf

Author

N. J. A. Sloane, Dec 22 2011

Extensions

More terms from Michel Marcus, Jun 03 2020

Status

approved