A259485 -id:A259485 - cp's OEIS frontend

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

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

nonn

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

			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

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.

Cf. A259485, A259666.
Row sums of A259786.
Main diagonal (shifted) of A259841.
Column k=1 of A259844.

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

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 *)

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

A259666 Number of n X n prime Tesler matrices.

Original entry on oeis.org

1, 1, 3, 18, 181, 2788, 62590, 1989540, 87979661, 5349559222, 443306080232, 49679250634068, 7473835936432840, 1498682325685621140, 397803907069442925517, 138847938093177059278212, 63325340852730727078521540, 37513306417359729218973719474, 28701720575221087513434901774347

Offset: 1

Alejandro H. Morales, Jul 02 2015

nonn

Number of n X n upper triangular matrices A of nonnegative integers such that a_{1,i} + a_{2,i} + ... + a_{i-1,i} - a_{i,i} - a_{i,i+1} - ... - a_{i,n} = -1, whose simple graph G with vertices 1,2,3..,n and edges (i,j) if a_{i,j} > 0 is connected.

			Example: For n =3 the a(3) = 3 matrices are [[0,1,0],[0,1,1],[0,0,2]], [[0,1,0],[0,0,2],[0,0,3]], [[0,0,1],[0,0,1],[0,0,3]].
E.g.f.: 1 + x+(1/2)*x^2+(3/6)*x^3+(18/24)*x^4+(181/120)*x^5+(2788/720)*x^6 + ...

A. Garsia and J. Haglund, A polynomial expression for the character of diagonal harmonics, Ann. Comb., to appear, 2015.

Cf. A008608, A259485.

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):
sa := 1 + log(1+ add(b(n)*x^n/n!,n=1..7)):
a := n -> n!*coeff(series(sa,x,n+1),x,n):
seq(a(i),i=1..6);

b[n_, i_, l_] := 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]];
c[n_] := b[1, n-1, Array[0&, n-1]];
a[n_] := a[n] = SeriesCoefficient[1 + Log[1 + Sum[c[k] x^k/k!, {k, 1, n}]], {x, 0, n}] n!;
Table[Print[n, " ", a[n]]; a[n], {n, 1, 19}] (* Jean-François Alcover, Nov 14 2020, after Alois P. Heinz in A008608 *)

E.g.f.: 1 + log( 1+ sum(n>=1, A008608(n) * x^n / n! ) ).

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

cp's OEIS Frontend

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.

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