login
A265163
Array of basis permutations, seen as a triangle read by rows: Row k (k >= 0) gives the values of b(n, k) = number of permutations of size n (2 <= n <= 2(k+1)) in the permutation basis B(k) (see Comments for further details).
3
1, 0, 2, 1, 0, 0, 6, 8, 1, 0, 0, 0, 24, 58, 18, 1, 0, 0, 0, 0, 120, 444, 244, 32, 1, 0, 0, 0, 0, 0, 720, 3708, 3104, 700, 50, 1, 0, 0, 0, 0, 0, 0, 5040, 33984, 39708, 13400, 1610, 72, 1, 0, 0, 0, 0, 0, 0, 0, 40320, 341136, 525240, 244708, 43320, 3206, 98, 1
OFFSET
0,3
COMMENTS
A right-jump in a permutation consists of taking an element and moving it somewhere to its right.
The set P(k) of permutations reachable from the identity after at most k right-jumps is a permutation-pattern avoiding set: it coincides with the set of permutation avoiding a set of patterns.
We define B(k) to be the smallest such set of "forbidden patterns" (the permutation pattern community calls such a set a "basis" for P(k), and its elements can be referred to as "right-jump basis permutations").
The number b(n,k) of permutations of size n in B(k) is given by the array in the present sequence.
The row sums give the sequence A265164 (i.e. this counts the permutations of any size in the basis B(k)).
The column sums give the sequence A265165 (i.e. this counts the permutations of size n in any B(k)).
LINKS
Cyril Banderier, Jean-Luc Baril, CĂ©line Moreira Dos Santos, Right jumps in permutations, Permutation Patterns 2015.
EXAMPLE
The number b(n, k) of basis permutations of length n where 2<=n<=11.
k\n | 2 3 4 5 6 7 8 9 10 11 | #B_k
0 | 1 | 1
1 | 0 2 1 | 3
2 | 0 0 6 8 1 | 15
3 | 0 0 0 24 58 18 1 | 101
4 | 0 0 0 0 120 444 244 32 1 | 841
5 | 0 0 0 0 0 720 3708 3104 700 50 | 8232
6 | 0 0 0 0 0 0 5040 33984 39708 13400 | 78732
----+--------------------------------------------------+------
Sum | 1 2 7 32 179 1182 8993 77440 744425 7901410 |
----+--------------------------------------------------+------
CROSSREFS
Cf. A265164 (row sums B(k)), A265165 (column sums).
Sequence in context: A371568 A267163 A357885 * A057275 A057271 A021480
KEYWORD
nonn,tabf
AUTHOR
Cyril Banderier, Dec 07 2015, with additional comments added Feb 06 2017.
STATUS
approved