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, 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

Author

Jean-François Alcover, Mar 11 2015

Keywords

• nonn
• tabl
• easy

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

Examples

			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.
		

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.

Crossrefs

Cf. A000217, A000260, A070914 (generalized Catalan numbers giving the number of paths), A103220.

Programs

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

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

Formula

a(n,m) = ((m + 1)/(n*(m*n + 1)))*binomial((m + 1)^2*n + m, n - 1).