cp's OEIS Frontend

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.

%I A255918 #32 Jun 09 2021 16:13:46
%S A255918 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,
%T A255918 9729,2530,1,1,28,791,13390,91881,146916,16965,1,1,36,1428,38591,
%U A255918 524256,2509584,2359968,118668,1,1,45,2388,94738,2180262,22533126
%N 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.
%C A255918 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
%C A255918 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
%H A255918 Olivier Bernardi and Nicolas Bonichon, <a href="http://dx.doi.org/10.1016/j.jcta.2008.05.005">Intervals in Catalan lattices and realizers of triangulations</a>, Journal of Combinatorial Theory, Series A 116:1 (2009), pp. 55-75.
%H A255918 M. Bousquet-Mélou, E. Fusy, and L.-F. Préville Ratelle, <a href="http://arxiv.org/abs/1106.1498">The number of intervals in the m-Tamari lattices</a>, arXiv:1106.1498 [math.CO], The Electronic Journal of Combinatorics 18, 2 (2011) P31.
%H A255918 Clay Cordova and Shu-Heng Shao, <a href="https://arxiv.org/abs/1502.08050">Counting Trees in Supersymmetric Quantum Mechanics</a> arXiv:1502.08050v2 [hep-th], 2015.
%H A255918 Thorsten Weist, <a href="https://arxiv.org/abs/0903.5442">Localization in quiver moduli spaces</a>, arXiv:0903.5442 [math.RT], 2009.
%F A255918 a(n,m) = ((m + 1)/(n*(m*n + 1)))*binomial((m + 1)^2*n + m, n - 1).
%e A255918 Array begins:
%e A255918 1,   1,    1,     1,      1,       1,       1,        1,        1, ...
%e A255918 1,   3,    6,    10,     15,      21,      28,       36,       45, ...
%e A255918 1,  13,   58,   170,    395,     791,    1428,     2388,     3765, ...
%e A255918 1,  68,  703,  3685,  13390,   38591,   94738,   206718,   412095, ...
%e A255918 1, 399, 9729, 91881, 524256, 2180262, 7291550, 20787390, 52450587, ...
%e A255918 ...
%e A255918 2nd row is A000217 (triangular numbers);
%e A255918 3rd row is A103220;
%e A255918 4th row is not in the OEIS;
%e A255918 2nd column is A000260 (number of intervals in the usual Tamari lattice of size n);
%e A255918 3rd column is not in the OEIS.
%t A255918 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
%o A255918 (Sage)
%o A255918 def a(n, m):
%o A255918     return (m + 1) * binomial((m + 1)**2 * n + m, n - 1) // (n*(m*n + 1)) # _F. Chapoton_, Mar 24 2021
%Y A255918 Cf. A000217, A000260, A070914 (generalized Catalan numbers giving the number of paths), A103220.
%K A255918 nonn,tabl,easy
%O A255918 1,5
%A A255918 _Jean-François Alcover_, Mar 11 2015