A008608 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.

1, 2, 7, 40, 357, 4820, 96030, 2766572, 113300265, 6499477726, 515564231770, 55908184737696, 8203615387086224, 1613808957720017838, 422045413500096791377, 145606442599303799948900, 65801956684134601408784992, 38698135339344702725297294600, 29437141738828506134939056167071, 28800381656420765181010517468370560

Offset: 1

Author

Glenn P. Tesler (gptesler(AT)euclid.ucsd.edu)

Keywords

• nonn

Comments

Garsia and Haglund call these Tesler matrices. - N. J. A. Sloane, Jul 04 2014

This is also the value of the type A_n Kostant partition function evaluated at (1,1,...,1,-n) in ZZ^(n+1). This is the number of ways of writing the vector (1,1,...,1,-n) in ZZ^(n+1) as a linear combination with nonnegative integer coefficients of the vectors e_i - e_j, for 1 <= iAlejandro H. Morales, Mar 11 2014

Examples

			For n = 3 there are seven matrices: [[1,0,0],[0,1,0],[0,0,1]], [[1,0,0],[0,0,1],[0,0,2]], [[0,0,1],[0,1,0],[0,0,2]], [[0,0,1],[0,0,1],[0,0,3]], [[0,1,0],[0,2,0],[0,0,1]], [[0,1,0],[0,1,1],[0,0,2]], [[0,1,0],[0,0,2],[0,0,3]], so a(3) = 7. - _Alejandro H. Morales_, Jul 03 2015
		

Links

• Joel B. Lewis, Table of n, a(n) for n = 1..26 (first 23 terms from Jay Pantone)
• 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.
• W. Baldoni and M. Vergne, Kostant partitions functions and flow polytopes, Transform. Groups, 13(3-4):447-469, 2008.
• J. Haglund, A polynomial expression for the Hilbert series of the quotient ring of diagonal coinvariants.
• A. Garsia and J. Haglund, A polynomial expression for the character of diagonal harmonics, Ann. Comb., 19 (2015), 693-703. See also the preprint.
• Ricky I. Liu, K. Mészáros, and A. H. Morales, Flow polytopes and the space of diagonal harmonics, arXiv preprint arXiv:1610.08370 [math.CO], 2016.
• K. Mészáros, A. H. Morales, and B. Rhoades, The polytope of Tesler matrices, arXiv preprint arXiv:1409.8566 [math.CO], 2014.
• Jason O'Neill, On the poset and asymptotics of Tesler Matrices, arXiv:1702.00866 [math.CO], 2017.
• Michèle Vergne et al., Maple programs for efficient computation of the Kostant partition function.

Crossrefs

Cf. A259485, A259666.

Row sums of A259786.

Main diagonal (shifted) of A259841.

Column k=1 of A259844.

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):
a := n -> multcoeff(n+1,F(n+1),[seq(1,i=1..n),-n],n+2):
seq(a(i),i=2..7) # Alejandro H. Morales, Mar 11 2014, Jun 28 2015
# second Maple program:
b:= proc(n, i, l) option remember; (m-> `if`(m=0, 1,
      `if`(i=0, b(l[1]+1, m-1, subsop(1=NULL, l)), add(
      b(n-j, i-1, subsop(i=l[i]+j, l)), j=0..n))))(nops(l))
    end:
a:= n-> b(1, n-1, [0$(n-1)]):
seq(a(n), n=1..14);  # Alois P. Heinz, Jul 05 2015

Mathematica

b[n_, i_, l_List] := b[n, i, l] = Function[{m}, 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}]]]][Length[l]]; a[n_] := b[1, n-1, Array[0&, n-1]]; Table[a[n], {n, 1, 14}] (* Jean-François Alcover, Jul 16 2015, after Alois P. Heinz *)

Extensions

a(7)-a(13) from Alejandro H. Morales, Mar 12 2014

a(14) from Alejandro H. Morales, Jun 04 2015

a(15)-a(22) from Alois P. Heinz, Jul 05 2015