A259485 Number of n X n connected Tesler matrices.

1, 1, 4, 27, 275, 4066, 85888, 2567269, 107630237, 6269269823, 502429080919, 54869692738326, 8091237358339821, 1597342350434681954, 418809228874760212806, 144760685900877097431589, 65510311668753649557469187, 38566383210089506976493649269, 29359678772700284486457832056879

Offset: 1

Author

Alejandro H. Morales, Jun 28 2015

Keywords

• nonn

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.

Examples

			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]].
		

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.

Crossrefs

Cf. A008608, A259666.

Programs

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 *)

Formula

a(n) = A008608(n) - Sum_{i=1..n-1} A008608(n-i)*a(i).

Extensions

a(15)-a(19) from Alois P. Heinz, Jul 05 2015