OFFSET
0,3
COMMENTS
An ordered pair of n-permutations ((a_1,a_2,...,a_n),(b_1,b_2,...,b_n)) has a common double descent at position i, 1<=i<=n-2, if a_i > a_i+1 > a_i+2 and b_i > b_i+1 > b_i+2.
REFERENCES
R. P. Stanley, Enumerative Combinatorics, Volume I, Second Edition, example 3.18.3e, page 366.
LINKS
Alois P. Heinz, Rows n = 0..60, flattened
P. Flajolet and R. Sedgewick, Analytic Combinatorics, 2009; page 209.
EXAMPLE
T(4,1) = 30: There are 9 such ordered pairs formed from the permutations 3421,2431,1432. There are 9 such ordered pairs formed from the permutations 4312,4213,3214. Then pairing each of these 6 permutations with 4321 gives 12 more ordered pairs with exactly 1 common double descent. 9+9+12 = 30.
Triangle T(n,k) begins:
1;
1;
4;
35, 1;
545, 30, 1;
13250, 1101, 48, 1;
463899, 51474, 2956, 70, 1;
...
MAPLE
b:= proc(n, u, v, t) option remember; expand(`if`(n=0, 1,
add(add(b(n-1, u-j, v-i, x)*t, i=1..v)+
add(b(n-1, u-j, v+i-1, 1), i=1..n-v), j=1..u)+
add(add(b(n-1, u+j-1, v-i, 1), i=1..v)+
add(b(n-1, u+j-1, v+i-1, 1), i=1..n-v), j=1..n-u)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2, 1)):
seq(T(n), n=0..10); # Alois P. Heinz, Apr 26 2020
MATHEMATICA
nn = 8; a = Apply[Plus, Table[Normal[Series[y x^3/(1 - y x - y x^2), {x, 0, nn}]][[n]]/(n +2)!^2, {n, 1, nn - 2}]] /. y -> y - 1; Map[Select[#, # > 0 &] &,
Range[0, nn]!^2 CoefficientList[Series[1/(1 - x - a), {x, 0, nn}], {x, y}]] // Grid
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Geoffrey Critzer, Apr 26 2020
STATUS
approved