The applets below visualize Hasse diagrams of the posets defined by level sets of certain tropical polynomials on the set of (n+1)-tuples of non-negative integers (a0, a1, ... , an). There are k such polynomials with values e1, e2, ... , ek, where k = floor(n/2). The first scrollbar controls the degree e0 = a0 + a1 + ... + an. The next scrollbars control the values of e1, ... , ek, and allow you to move the diagram by scalings and translations. Note: for all indices i, we must have e(i) + e(i+2) greater than or equal to 2 e(i+1) for the poset to be non-empty.

See the article on Tropical decomposition of Young's partition lattice for details.

n = 3:

      e1 = min(a0+a1,a0+a3,a2+a3)

      Applet for n = 3

n = 4:

      e1 = min(a0+a1+a2,a0+a1+a4,a0+a3+a4,a2+a3+a4)

      e2 = min(a0,a2,a4)

      Applet for n = 4

n = 5:

      e1 = min(a0+a1+a2+a3,a0+a1+a2+a5,a0+a1+a4+a5,a0+a3+a4+a5,a2+a3+a4+a5)

      e2 = min(a0+a1,a0+a3,a0+a5,a2+a3,a2+a5,a4+a5)

      Applet for n = 5

n = 6:

      e1 = min(a0+a1+a2+a3+a4,a0+a1+a2+a3+a6,a0+a1+a2+a5+a6,a0+a1+a4+a5+a6,a0+a3+a4+a5+a6,a2+a3+a4+a5+a6)

      e2 = min(a0+a1+a2,a0+a1+a4,a0+a1+a6,a0+a3+a4,a0+a3+a6,a0+a5+a6,a2+a3+a4,a2+a3+a6,a2+a5+a6,a4+a5+a6)

      e3 = min(a0,a2,a4,a6)

      Applet for n = 6

n = 7:

      e1 = min(0+a1+a2+a3+a4+a5,0+a1+a2+a3+a4+a7,0+a1+a2+a3+a6+a7,0+a1+a2+a5+a6+a7,0+a1+a4+a5+a6+a7,1+a3+a4+a5+a6+a7,2+a3+a4+a5+a6+a7)

      e2 = min(a0+a1+a2+a3,a0+a1+a2+a5,a0+a1+a2+a7,a0+a1+a4+a5,a0+a1+a4+a7,a0+a1+a6+a7,a0+a3+a4+a5,a0+a3+a4+a7,a0+a3+a6+a7,a0+a5+a6+a7,a2+a3+a4+a5,a2+a3+a4+a7,a2+a3+a6+a7,a2+a5+a6+a7,a4+a5+a6+a7)

      e3 = min(a0+a1,a0+a3,a0+a5,a0+a7,a2+a3,a2+a5,a2+a7,a4+a5,a4+a7,a6+a7)

      Applet for n = 7