let's return to the down-sets in the "interval containment" order on 12,13,14,23,24,34:

[],[12],[23],[34],[12 23],[12 34],[23 34],[12 23 34],[13],[13 34],[24],[12 24],[13 24],[14]

i want to see the cardinalities of the down-sets now, so instead of just listing the generators of the down-set i'll list all the elements:

[],[12],[23],[34],[12 23],[12 34],[23 34],[12 23 34],[13 12 23],[13 12 23 34],[24 23 34],[12 24 23 34],[13 12 23 24 34],[14 13 24 12 23 34]

so it looks like we have

1 0
3 1's
3 2's
3 3's
2 4's
1 5
1 6

let's compare that to the coefficients of (1)(1+x)(1+x+x^2)(1+x+x^2+x^3) (which gives the level sizes in the kaleidoscope group 4!):

1 0
3 1's
5 2's
6 3's
5 4's
3 5's
1 6

1234

2134 - 1324 - 1243

2314 3124 - 2143 - 1342 1423

3214 4123 - 2341 2413 - 3142 1432 ??????

3241 2431 4213 - 3412 4132 ??????

3421 4231 4312

4321

??are we dealing here with a murphy congruence maybe?? perhaps not...

we're tentatively assuming that none of the schubert varieties here are singular...