This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A265163 #55 Jul 09 2025 04:40:27 %S A265163 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, %T A265163 0,720,3708,3104,700,50,1,0,0,0,0,0,0,5040,33984,39708,13400,1610,72, %U A265163 1,0,0,0,0,0,0,0,40320,341136,525240,244708,43320,3206,98,1 %N 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). %C A265163 A right-jump in a permutation consists of taking an element and moving it somewhere to its right. %C A265163 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. %C A265163 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"). %C A265163 The number b(n,k) of permutations of size n in B(k) is given by the array in the present sequence. %C A265163 The row sums give the sequence A265164 (i.e. this counts the permutations of any size in the basis B(k)). %C A265163 The column sums give the sequence A265165 (i.e. this counts the permutations of size n in any B(k)). %H A265163 Cyril Banderier, Jean-Luc Baril, Céline Moreira Dos Santos, <a href="https://lipn.univ-paris13.fr/~banderier/Papers/rightjump.pdf">Right jumps in permutations</a>, Permutation Patterns 2015. %e A265163 The number b(n, k) of basis permutations of length n where 2<=n<=11. %e A265163 k\n | 2 3 4 5 6 7 8 9 10 11 | #B_k %e A265163 0 | 1 | 1 %e A265163 1 | 0 2 1 | 3 %e A265163 2 | 0 0 6 8 1 | 15 %e A265163 3 | 0 0 0 24 58 18 1 | 101 %e A265163 4 | 0 0 0 0 120 444 244 32 1 | 841 %e A265163 5 | 0 0 0 0 0 720 3708 3104 700 50 | 8232 %e A265163 6 | 0 0 0 0 0 0 5040 33984 39708 13400 | 78732 %e A265163 ----+--------------------------------------------------+------ %e A265163 Sum | 1 2 7 32 179 1182 8993 77440 744425 7901410 | %e A265163 ----+--------------------------------------------------+------ %Y A265163 Cf. A265164 (row sums B(k)), A265165 (column sums). %K A265163 nonn,tabf %O A265163 0,3 %A A265163 _Cyril Banderier_, Dec 07 2015, with additional comments added Feb 06 2017