A259841 -id:A259841 - 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

A259787 Total element sum of all n X n Tesler matrices of nonnegative integers.

Original entry on oeis.org

1, 5, 31, 270, 3370, 60146, 1522031, 54055976, 2666453502, 180847717069, 16704822358932, 2082808024263350, 347639192485104658, 77076883307827211845, 22537752778732740525833, 8633258320969387044105210, 4305220991520242104331411368, 2778601200692503839128415662124

Offset: 1

Alois P. Heinz, Jul 05 2015

nonn

For the definition of Tesler matrices see A008608.

			There are two 2 X 2 Tesler matrices: [1,0; 0,1], [0,1; 0,2], the total sum of all elements gives a(2) = 5.

Alois P. Heinz, Table of n, a(n) for n = 1..21

Cf. A008608, A259786, A259841.

b:= proc(n, i, l) option remember; (m-> `if`(m=0, [1, 0], `if`(i=0,
      (p-> p+[0, p[1]*(l[1]+1)])(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-> (p-> p[1]+p[2])(b(1, n-1, [0$(n-1)])):
seq(a(n), n=1..14);

b[n_, i_, l_] := b[n, i, l] = Function[m, If[m == 0, {1, 0}, If[i == 0, Function[p, p + {0, p[[1]]*(l[[1]] + 1)}][b[l[[1]] + 1, m - 1, ReplacePart[l, 1 -> Nothing]]], Sum[b[n - j, i - 1, ReplacePart[l, i -> l[[i]] + j]], {j, 0, n}]]]][Length[l]];
a[n_] := Function[p, p[[1]] + p[[2]]][b[1, n - 1, Table[0, {n - 1}]]];
Table[a[n], {n, 1, 14}] (* Jean-François Alcover, Jun 27 2022, after Alois P. Heinz *)

a(n) = Sum_{k=0..n*(n-1)/2} (n+k) * A259786(n,k).

a(n) = Sum_{k=0..n} k * A259841(n,k).

A259842 Number of nonzero elements in all n X n Tesler matrices of nonnegative integers.

Original entry on oeis.org

1, 4, 22, 178, 2114, 36398, 896128, 31136246, 1508259823, 100727634758, 9179951931947, 1131033520118692, 186769092227016256, 41008206412935719870, 11884278052476825052541, 4514826724675651497522250, 2234142899928806917974566378, 1431533853656098851281985968328

Offset: 1

Alois P. Heinz, Jul 06 2015

nonn

For the definition of Tesler matrices see A008608.

			There are two 2 X 2 Tesler matrices: [1,0; 0,1], [0,1; 0,2], containing four nonzero elements, thus a(2) = 4.

Row sums of A259841.

Cf. A008608.

g:= u-> `if`(u=0, 0, 1):
b:= proc(n, i, l) option remember; (m->`if`(m=0, [1, g(n)], `if`(i=0,
     (p->p+[0, p[1]*g(n)])(b(l[1]+1, m-1, subsop(1=NULL, l))), add(
     (p->p+[0, p[1]*g(j)])(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)])[2]:
seq(a(n), n=1..14);

g[u_] := If[u == 0, 0, 1];
b[n_, i_, l_] := b[n, i, l] = Function[m, If[m == 0, {1, g[n]}, If[i == 0,
     # + {0, #[[1]] g[n]}&[b[l[[1]] + 1, m - 1, ReplacePart[l, 1 ->
     Nothing]]], Sum[# + {0, #[[1]] g[j]}&[b[n - j, i - 1, ReplacePart[
     l, i -> l[[i]] + j]]], {j, 0, n}]]]][Length[l]];
a[n_] := b[1, n - 1, Table[0, {n - 1}]][[2]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 18}] (* Jean-François Alcover, May 15 2022, after Alois P. Heinz *)

a(n) = Sum_{k=1..n} A259841(n,k).

A259843 Number of 1 elements in all n X n Tesler matrices of nonnegative integers.

Original entry on oeis.org

1, 3, 15, 117, 1367, 23329, 570933, 19740068, 951916938, 63295826369, 5743788894259, 704672958229270, 115877288304781885, 25338423080304873558, 7313716095786704678585, 2767636572798780219442327, 1364367542961142350256304582, 871016593387715393187604249892

Offset: 1

Alois P. Heinz, Jul 06 2015

nonn

For the definition of Tesler matrices see A008608.

			There are two 2 X 2 Tesler matrices: [1,0; 0,1], [0,1; 0,2], containing three 1's, thus a(2) = 3.

Column k=1 of A259841.

g:= u-> `if`(u=1, 1, 0):
b:= proc(n, i, l) option remember; (m->`if`(m=0, [1, g(n)], `if`(i=0,
     (p->p+[0, p[1]*g(n)])(b(l[1]+1, m-1, subsop(1=NULL, l))), add(
     (p->p+[0, p[1]*g(j)])(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)])[2]:
seq(a(n), n=1..14);

g[u_] := If[u == 1, 1, 0];
b[n_, i_, l_] := b[n, i, l] = Function[m, If[m == 0, {1, g[n]}, If[i == 0, # + {0, #[[1]] g[n]}&[b[l[[1]] + 1, m - 1, ReplacePart[l, 1 -> Nothing]] ], Sum[# + {0, #[[1]] g[j]}&[b[n - j, i - 1, ReplacePart[l, i -> l[[i]] + j]]], {j, 0, n}]]]][Length[l]];
a[n_] := b[1, n - 1, Table[0, {n - 1}]][[2]];
Table[Print[n, " ", a[n]]; a[n], {n, 1, 18}] (* Jean-François Alcover, May 15 2022, after Alois P. Heinz *)

a(n) = A259841(n,1).

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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A259787 Total element sum of all n X n Tesler matrices of nonnegative integers.

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A259842 Number of nonzero elements in all n X n Tesler matrices of nonnegative integers.

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A259843 Number of 1 elements in all n X n Tesler matrices of nonnegative integers.

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Maple

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