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A255918
Array a(n,m) read by descending antidiagonals giving the number of intervals in a generalized Tamari lattice of m-ballot paths of size n.
1
1, 1, 1, 1, 3, 1, 1, 6, 13, 1, 1, 10, 58, 68, 1, 1, 15, 170, 703, 399, 1, 1, 21, 395, 3685, 9729, 2530, 1, 1, 28, 791, 13390, 91881, 146916, 16965, 1, 1, 36, 1428, 38591, 524256, 2509584, 2359968, 118668, 1, 1, 45, 2388, 94738, 2180262, 22533126
OFFSET
1,5
COMMENTS
This array occurs in counting the degeneracies in the supersymmetric ground states of the Kronecker model of quiver quantum mechanics. See Cordova and Shao, 1.4. - Peter Bala, Oct 29 2017
In mathematical terms, this corresponds to the homology of some moduli spaces of semi-stable representations of the Kronecker quiver K_m with dimension vector (n,n+1). F. Chapoton, Jun 09 2021
LINKS
Olivier Bernardi and Nicolas Bonichon, Intervals in Catalan lattices and realizers of triangulations, Journal of Combinatorial Theory, Series A 116:1 (2009), pp. 55-75.
M. Bousquet-Mélou, E. Fusy, and L.-F. Préville Ratelle, The number of intervals in the m-Tamari lattices, arXiv:1106.1498 [math.CO], The Electronic Journal of Combinatorics 18, 2 (2011) P31.
Clay Cordova and Shu-Heng Shao, Counting Trees in Supersymmetric Quantum Mechanics arXiv:1502.08050v2 [hep-th], 2015.
Thorsten Weist, Localization in quiver moduli spaces, arXiv:0903.5442 [math.RT], 2009.
FORMULA
a(n,m) = ((m + 1)/(n*(m*n + 1)))*binomial((m + 1)^2*n + m, n - 1).
EXAMPLE
Array begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, ...
1, 3, 6, 10, 15, 21, 28, 36, 45, ...
1, 13, 58, 170, 395, 791, 1428, 2388, 3765, ...
1, 68, 703, 3685, 13390, 38591, 94738, 206718, 412095, ...
1, 399, 9729, 91881, 524256, 2180262, 7291550, 20787390, 52450587, ...
...
2nd row is A000217 (triangular numbers);
3rd row is A103220;
4th row is not in the OEIS;
2nd column is A000260 (number of intervals in the usual Tamari lattice of size n);
3rd column is not in the OEIS.
MATHEMATICA
a[n_, m_] := ((m + 1)/(n*(m*n + 1)))*Binomial[(m + 1)^2*n + m, n - 1]; Table[a[n - m, m], {n, 1, 12}, {m, n - 1, 0, -1}] // Flatten
PROG
(Sage)
def a(n, m):
return (m + 1) * binomial((m + 1)**2 * n + m, n - 1) // (n*(m*n + 1)) # F. Chapoton, Mar 24 2021
CROSSREFS
Cf. A000217, A000260, A070914 (generalized Catalan numbers giving the number of paths), A103220.
Sequence in context: A235114 A272866 A228899 * A102479 A228902 A053193
KEYWORD
nonn,tabl,easy
AUTHOR
STATUS
approved