OFFSET
0,3
COMMENTS
Let ( (a_1,a_2,...,a_n), (b_1,b_2,...,b_n) be an ordered pair of n-permutations. Then the pairs (a_i,a_(i+1)) and (b_i,b_(i+1)) are both rises, both falls, a rise and a fall, or a fall and a rise. T(n,k) is the number of ordered pairs of n-permutations that have a total of k rise/falls and fall/rises.
LINKS
Alois P. Heinz, Rows n = 0..70, flattened
L. Carlitz, Richard Scoville, and Theresa Vaughan, Enumeration of pairs of permutations and sequences, Bull. Amer. Math. Soc. 80(5) (1974), 881-884.
FORMULA
Sum_{n>=0} Sum_{k=0..n-1} u^k*z^n/(n!)^2 = ((1 - u) A((1 - u) z))/(1 - u A((1 - u) z)) where A(z) = Sum_{n>=0} A060350*z^n/(n!)^2. Theorem 4 in Carlitz, Scoville, Vaughan link.
EXAMPLE
Triangle begins:
1;
1;
2, 2;
10, 16, 10;
88, 200, 200, 88;
1216, 3536, 4896, 3536, 1216;
...
In the ordered pair of permutations ( (1,2,3,5,4), (4,2,1,3,5) ) we have a rise/fall, rise/fall, rise/rise, fall/rise. So this ordered pair is counted in T(5,3).
MAPLE
b:= proc(n, u, v) option remember; expand(`if`(n=0, 1,
add(add(b(n-1, u-j, v-i), i=1..v)+
add(b(n-1, u-j, v+i-1)*x, i=1..n-v), j=1..u)+
add(add(b(n-1, u+j-1, v-i)*x, i=1..v)+
add(b(n-1, u+j-1, v+i-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)):
seq(T(n), n=0..10); # Alois P. Heinz, May 01 2023
MATHEMATICA
nn = 8; A[z_] := Total[Select[Import["https://oeis.org/A060350/b060350.txt", "Table"], Length@# == 2 &][[All, 2]]*Table[z^n/n!^2, {n, 0, 250}]]; B[n_] := n!^2; e[z_] := Sum[z^n/B[n], {n, 0, nn}]; Map[Select[#, # > 0 &] &, Table[B[n], {n, 0, nn}] CoefficientList[Series[((1 - u) A[(1 - u) z])/(1 - u A[(1 - u) z]), {z, 0, nn}], {z, u}]] // Flatten
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Geoffrey Critzer, May 01 2023
STATUS
approved