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A259485 Number of n X n connected Tesler matrices. 2
1, 1, 4, 27, 275, 4066, 85888, 2567269, 107630237, 6269269823, 502429080919, 54869692738326, 8091237358339821, 1597342350434681954, 418809228874760212806, 144760685900877097431589, 65510311668753649557469187, 38566383210089506976493649269, 29359678772700284486457832056879 (list; graph; refs; listen; history; text; internal format)
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.
FORMULA
a(n) = A008608(n) - Sum_{i=1..n-1} A008608(n-i)*a(i).
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
Sequence in context: A104653 A194787 A020558 * A362699 A193467 A357174
KEYWORD
nonn
AUTHOR
EXTENSIONS
a(15)-a(19) from Alois P. Heinz, Jul 05 2015
STATUS
approved

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Last modified July 23 23:47 EDT 2024. Contains 374575 sequences. (Running on oeis4.)