Some Interesting Probability Limit Theorems About Plane Partitions

                               of Bounded Height

                              by Shalosh B. Ekhad

                Theorem Number , 1,  : Consider the set of plane

                 partitions whose 3D Ferrers diagram is  inside

                            an n by n by , 1,  box

                    Let the r.v. be the: the number of cells

      The even, 2r-th, moment, about the mean is (asymptotically, ignoring

                               1
                              ----, and beyond):
                                3
                               n

               2        3 r
(2/3 n + 7/12 n  + 1/6 n )  (2 r)!

    /                                   2              \
    |    3 r (-1 + r)   r (-1 + r) (63 r  - 75 r + 248)|   /      r
    |1 - ------------ + -------------------------------|  /  (r! 2 )
    |        10 n                         2            | /
    \                               1400 n             /

                                                  2        3
                   The variance is, 2/3 n + 7/12 n  + 1/6 n

      After dividing by the variance to the power r (i.e. normalizing it)

                   we get that the normalized 2r-th moment is

                 /                                   2              \
                 |    3 r (-1 + r)   r (-1 + r) (63 r  - 75 r + 248)|
          (2 r)! |1 - ------------ + -------------------------------|
                 |        10 n                         2            |
                 \                               1400 n             /
          -----------------------------------------------------------
                                         r
                                     r! 2

            In particular, it follows that this discrete prob. dist.

                is asymptotically normal, but it does much more!

                              We have established

                         the asymptotic to, 3, terms .

              Of course, by symmetry, all the odd moments are zero

      but we can derive this directly, getting that the odd moments eqaul

                                       0

                Theorem Number , 2,  : Consider the set of plane

                 partitions whose 3D Ferrers diagram is  inside

                            an n by n by , 2,  box

                    Let the r.v. be the: the number of cells

      The even, 2r-th, moment, about the mean is (asymptotically, ignoring

                               1
                              ----, and beyond):
                                3
                               n

              2        3 r
(5/3 n + 4/3 n  + 1/3 n )  (2 r)!

    /                                   2              \
    |    3 r (-1 + r)   r (-1 + r) (63 r  - 75 r + 458)|   /      r
    |1 - ------------ + -------------------------------|  /  (r! 2 )
    |        20 n                         2            | /
    \                               5600 n             /

                                                  2        3
                    The variance is, 5/3 n + 4/3 n  + 1/3 n

      After dividing by the variance to the power r (i.e. normalizing it)

                   we get that the normalized 2r-th moment is

                 /                                   2              \
                 |    3 r (-1 + r)   r (-1 + r) (63 r  - 75 r + 458)|
          (2 r)! |1 - ------------ + -------------------------------|
                 |        20 n                         2            |
                 \                               5600 n             /
          -----------------------------------------------------------
                                         r
                                     r! 2

            In particular, it follows that this discrete prob. dist.

                is asymptotically normal, but it does much more!

                              We have established

                         the asymptotic to, 3, terms .

              Of course, by symmetry, all the odd moments are zero

      but we can derive this directly, getting that the odd moments eqaul

                                       0

                Theorem Number , 3,  : Consider the set of plane

                 partitions whose 3D Ferrers diagram is  inside

                            an n by n by , 3,  box

                    Let the r.v. be the: the number of cells

      The even, 2r-th, moment, about the mean is (asymptotically, ignoring

                               1
                              ----, and beyond):
                                3
                               n

            2        3 r
(3 n + 9/4 n  + 1/2 n )  (2 r)!

    /                                 2              \
    |    r (-1 + r)   r (-1 + r) (21 r  - 25 r + 176)|   /      r
    |1 - ---------- + -------------------------------|  /  (r! 2 )
    |       10 n                        2            | /
    \                             4200 n             /

                                                 2        3
                     The variance is, 3 n + 9/4 n  + 1/2 n

      After dividing by the variance to the power r (i.e. normalizing it)

                   we get that the normalized 2r-th moment is

                  /                                 2              \
                  |    r (-1 + r)   r (-1 + r) (21 r  - 25 r + 176)|
           (2 r)! |1 - ---------- + -------------------------------|
                  |       10 n                        2            |
                  \                             4200 n             /
           ---------------------------------------------------------
                                         r
                                     r! 2

            In particular, it follows that this discrete prob. dist.

                is asymptotically normal, but it does much more!

                              We have established

                         the asymptotic to, 3, terms .

              Of course, by symmetry, all the odd moments are zero

      but we can derive this directly, getting that the odd moments eqaul

                                       0

                Theorem Number , 4,  : Consider the set of plane

                 partitions whose 3D Ferrers diagram is  inside

                            an n by n by , 4,  box

                    Let the r.v. be the: the number of cells

      The even, 2r-th, moment, about the mean is (asymptotically, ignoring

                               1
                              ----, and beyond):
                                3
                               n

                2        3 r
(14/3 n + 10/3 n  + 2/3 n )  (2 r)!

    /                                   2              \
    |    3 r (-1 + r)   r (-1 + r) (63 r  - 75 r + 458)|   /      r
    |1 - ------------ + -------------------------------|  /  (r! 2 )
    |        40 n                         2            | /
    \                              22400 n             /

                                                   2        3
                   The variance is, 14/3 n + 10/3 n  + 2/3 n

      After dividing by the variance to the power r (i.e. normalizing it)

                   we get that the normalized 2r-th moment is

                 /                                   2              \
                 |    3 r (-1 + r)   r (-1 + r) (63 r  - 75 r + 458)|
          (2 r)! |1 - ------------ + -------------------------------|
                 |        40 n                         2            |
                 \                              22400 n             /
          -----------------------------------------------------------
                                         r
                                     r! 2

            In particular, it follows that this discrete prob. dist.

                is asymptotically normal, but it does much more!

                              We have established

                         the asymptotic to, 3, terms .

              Of course, by symmetry, all the odd moments are zero

      but we can derive this directly, getting that the odd moments eqaul

                                       0

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