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BettiBounds :: gradedBettiChampion

gradedBettiChampion -- Compute Betti tables of the examples with maximal total Betti numbers

Synopsis

Description

Computes the Betti table of a simplicial complex of dimension q on q+c variables with maximal total Betti numbers bounds among all complexes obtained by a sequence of c-1 stellar subdivisions from the boundary complex of any simplex.

By putting the additional argument i one can obtain the Betti table of the i-th intermediate complex.

For the construction of the simplicial complexes see Section 5 of

[6] J. Boehm, S. Papadakis: Bounds on the Betti numbers of successive stellar subdivisions of a simplex, http://arxiv.org/abs/1212.4358

i1 : gradedBettiChampion 3

            0 1 2 3
o1 = total: 1 5 5 1
         0: 1 . . .
         1: . 1 . .
         2: . 4 4 .
         3: . . 1 .
         4: . . . 1

o1 : BettiTally
i2 : gradedBettiChampion 4

            0  1  2  3 4
o2 = total: 1 11 20 11 1
         0: 1  .  .  . .
         1: .  .  .  . .
         2: .  1  .  . .
         3: .  9  9  1 .
         4: .  .  2  . .
         5: .  1  9  9 .
         6: .  .  .  1 .
         7: .  .  .  . .
         8: .  .  .  . 1

o2 : BettiTally
i3 : gradedBettiChampion 5

            0  1  2  3  4 5
o3 = total: 1 23 62 62 23 1
         0: 1  .  .  .  . .
         1: .  .  .  .  . .
         2: .  .  .  .  . .
         3: .  1  .  .  . .
         4: . 18 22  7  1 .
         5: .  .  2  .  . .
         6: .  3 17 14  . .
         7: .  . 14 17  3 .
         8: .  .  .  2  . .
         9: .  1  7 22 18 .
        10: .  .  .  .  1 .
        11: .  .  .  .  . .
        12: .  .  .  .  . .
        13: .  .  .  .  . 1

o3 : BettiTally
i4 : gradedBettiChampion 6

            0  1   2   3   4  5 6
o4 = total: 1 47 170 248 170 47 1
         0: 1  .   .   .   .  . .
         1: .  .   .   .   .  . .
         2: .  .   .   .   .  . .
         3: .  .   .   .   .  . .
         4: .  1   .   .   .  . .
         5: . 35  55  31   9  1 .
         6: .  .   2   .   .  . .
         7: .  6  26  20   .  . .
         8: .  .  21  26   6  . .
         9: .  .  30  47  21  4 .
        10: .  4  21  47  30  . .
        11: .  .   6  26  21  . .
        12: .  .   .  20  26  6 .
        13: .  .   .   .   2  . .
        14: .  1   9  31  55 35 .
        15: .  .   .   .   .  1 .
        16: .  .   .   .   .  . .
        17: .  .   .   .   .  . .
        18: .  .   .   .   .  . .
        19: .  .   .   .   .  . 1

o4 : BettiTally
i5 : gradedBettiChampion 10

            0   1    2     3     4     5     6     7    8   9 10
o5 = total: 1 767 5762 19388 37928 47072 37928 19388 5762 767  1
         0: 1   .    .     .     .     .     .     .    .   .  .
         1: .   .    .     .     .     .     .     .    .   .  .
         2: .   .    .     .     .     .     .     .    .   .  .
         3: .   .    .     .     .     .     .     .    .   .  .
         4: .   .    .     .     .     .     .     .    .   .  .
         5: .   .    .     .     .     .     .     .    .   .  .
         6: .   .    .     .     .     .     .     .    .   .  .
         7: .   .    .     .     .     .     .     .    .   .  .
         8: .   1    .     .     .     .     .     .    .   .  .
         9: . 519 1803  2815  2561  1471   545   127   17   1  .
        10: .   .    2     .     .     .     .     .    .   .  .
        11: .  28   82    54     .     .     .     .    .   .  .
        12: .   .   55    82    28     .     .     .    .   .  .
        13: .   .  139   299   217    56     .     .    .   .  .
        14: .  56  482  1016   895   371    70     .    .   .  .
        15: .   .  419  1305  1652  1091   385    56    .   .  .
        16: .   .  503  1737  2489  1959   925   245   28   .  .
        17: .   .  538  1906  2869  2463  1345   461   91   8  .
        18: .  70  399  1034  1152   509    56     .    .   .  .
        19: .   .   84   493   734   384    56     .    .   .  .
        20: .   .   56   703  1479  1195   427    70    .   .  .
        21: .   .   56   649  1453  1249   455    70    .   .  .
        22: .   .   56   928  2457  2685  1490   455   56   .  .
        23: .  56  385  1538  3095  3093  1537   385   56   .  .
        24: .   .   70   901  2769  3643  2469   981  245  28  .
        25: .   .   70   455  1306  1535   676    56    .   .  .
        26: .   .   70   455  1416  1832   920   112    .   .  .
        27: .   .    .   112   920  1832  1416   455   70   .  .
        28: .   .    .    56   676  1535  1306   455   70   .  .
        29: .  28  245   981  2469  3643  2769   901   70   .  .
        30: .   .   56   385  1537  3093  3095  1538  385  56  .
        31: .   .   56   455  1490  2685  2457   928   56   .  .
        32: .   .    .    70   455  1249  1453   649   56   .  .
        33: .   .    .    70   427  1195  1479   703   56   .  .
        34: .   .    .     .    56   384   734   493   84   .  .
        35: .   .    .     .    56   509  1152  1034  399  70  .
        36: .   8   91   461  1345  2463  2869  1906  538   .  .
        37: .   .   28   245   925  1959  2489  1737  503   .  .
        38: .   .    .    56   385  1091  1652  1305  419   .  .
        39: .   .    .     .    70   371   895  1016  482  56  .
        40: .   .    .     .     .    56   217   299  139   .  .
        41: .   .    .     .     .     .    28    82   55   .  .
        42: .   .    .     .     .     .     .    54   82  28  .
        43: .   .    .     .     .     .     .     .    2   .  .
        44: .   1   17   127   545  1471  2561  2815 1803 519  .
        45: .   .    .     .     .     .     .     .    .   1  .
        46: .   .    .     .     .     .     .     .    .   .  .
        47: .   .    .     .     .     .     .     .    .   .  .
        48: .   .    .     .     .     .     .     .    .   .  .
        49: .   .    .     .     .     .     .     .    .   .  .
        50: .   .    .     .     .     .     .     .    .   .  .
        51: .   .    .     .     .     .     .     .    .   .  .
        52: .   .    .     .     .     .     .     .    .   .  .
        53: .   .    .     .     .     .     .     .    .   .  1

o5 : BettiTally

Intermediate steps:

i6 : gradedBettiChampion(3,1)

            0 1
o6 = total: 1 1
         0: 1 .
         1: . .
         2: . .
         3: . .
         4: . 1

o6 : BettiTally
i7 : gradedBettiChampion(3,2)

            0 1 2
o7 = total: 1 2 1
         0: 1 . .
         1: . . .
         2: . 2 .
         3: . . .
         4: . . 1

o7 : BettiTally
i8 : gradedBettiChampion(3,3)

            0 1 2 3
o8 = total: 1 5 5 1
         0: 1 . . .
         1: . 1 . .
         2: . 4 4 .
         3: . . 1 .
         4: . . . 1

o8 : BettiTally

See also

  • bettiBounds -- Compute the Betti bounds
  • champion -- Compute simplicial complexes with maximal total Betti numbers
  • cache -- Cache table

Ways to use gradedBettiChampion :

  • gradedBettiChampion(ZZ)
  • gradedBettiChampion(ZZ,ZZ)