OFFSET
1,3
COMMENTS
Number of n X n upper triangular matrices A of nonnegative integers such that a_1i + a_2i + ... + a_{i-1,i} - a_ii - a_{i,i+1} - ... - a_in = -1, with lowest lattice path above the positive entries not touching the diagonal.
LINKS
D. Armstrong, A. Garsia, J. Haglund, B. Rhoades and B. Sagan, Combinatorics of Tesler matrices in the theory of parking functions and diagonal harmonics, J. of Combin., 3(3):451-494, 2012.
D. Armstrong, Tesler Matrices, slides, Saganfest, March 2014.
EXAMPLE
For n = 3 the a(3) = 4 matrices are [[0,1,0],[0,1,1],[0,0,2]], [[0,1,0],[0,0,2],[0,0,3]], [[0,0,1],[0,1,0],[0,0,2]], [[0,0,1],[0,0,1],[0,0,3]].
MAPLE
multcoeff:=proc(n, f, coeffv, k)
local i, currcoeff;
currcoeff:=f;
for i from 1 to n do
currcoeff:=`if`(coeffv[i]=0, coeff(series(currcoeff, x[i], k), x[i], 0), coeff(series(currcoeff, x[i], k), x[i]^coeffv[i]));
end do;
return currcoeff;
end proc:
F:=n->mul(mul((1-x[i]*x[j]^(-1))^(-1), j=i+1..n), i=1..n):
b := n -> multcoeff(n+1, F(n+1), [seq(1, i=1..n), -n], n+2):
a := n -> `if`(n=1, 1, b(n)-add(b(n-i)*a(i), i=1..n-1)):
seq(a(i), i=2..6)
MATHEMATICA
b[n_, i_, l_] := b[n, i, l] = With[{m = Length[l]}, If[m == 0, 1, If[i == 0, b[l[[1]] + 1, m - 1, ReplacePart[l, 1 -> Sequence[]]], Sum[b[n - j, i - 1, ReplacePart[l, i -> l[[i]] + j]], {j, 0, n}]]]];
c[n_] := b[1, n - 1, Array[0&, n - 1]];
a[n_] := a[n] = c[n] - Sum[c[n - i] a[i], {i, 1, n - 1}];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 19}] (* Jean-François Alcover, Nov 13 2020, after Alois P. Heinz in A008608 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Alejandro H. Morales, Jun 28 2015
EXTENSIONS
a(15)-a(19) from Alois P. Heinz, Jul 05 2015
STATUS
approved