SmirnovWordCount

The words with each letter used this many times and no letter twice in a row: the Carlitz-Scoville-Vaughan sum over the numbers of runs of each letter, taken as a binomial convolution over the parts.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001097via IntegerPartition✓↗

SmirnovWordCount(partition)The words with each letter used this many times and no letter twice in a row: the Carlitz-Scoville-Vaughan sum over the numbers of runs of each letter, as a binomial convolution of the parts' signed compositions counts.

Typed signatures

Type	Package	Applies
(integer_partition) -> number	combinatorics	always

Domain: Partition statistics

Details

• Defined over IntegerPartition as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a IntegerPartition and nothing else — it reads values against their positions, or walks the orbits, so it needs the bijection. Applying it to a bare list is a type error, not a wrong answer.
• FindStat's St001097 reads it as a coefficient of the formal group law for linear orders; its description gives the Smirnov words.

Definition

Epsil, as the implementation runs it.

on IntegerPartition, over _x

1 if Length(_x) < 1 else Max(Sum(Fold(Function(Map(
                                                    Function(Sum(Map(Function((0 if j - 1 >
                                                                                    i - 1 else (1 if i - 1 ==
                                                                                                0 else Product(Range(1, i - 1))) /
                                                                                          ((1 if j - 1 ==
                                                                                                 0 else Product(Range(1, j - 1))) *
                                                                                           (1 if i - 1 -
                                                                                                 (j - 1) ==
                                                                                                 0 else Product(Range(1, i - 1 -
                                                                                                                        (j - 1)))))) *
                                                                              a[i - j +
                                                                                1] *
                                                                              (0 if j == 1 ||
                                                                                    j - 1 >
                                                                                    b else (-1) ^
                                                                                          (b -
                                                                                           (j - 1)) *
                                                                                          (0 if j - 2 >
                                                                                                b - 1 else (1 if b - 1 ==
                                                                                                            0 else Product(Range(1, b - 1))) /
                                                                                                      ((1 if j - 2 ==
                                                                                                             0 else Product(Range(1, j - 2))) *
                                                                                                       (1 if b - 1 -
                                                                                                             (j - 2) ==
                                                                                                             0 else Product(Range(1, b - 1 -
                                                                                                                                    (j - 2))))))), j), Range(1, i))), i),
                                                    Range(1, Sum(_x) + 1),
                                                  ), a, b), Map(
                                                                Function(
                                                                          1 if i ==
                                                                               1 else 0,
                                                                          i,
                                                                        ),
                                                                Range(
                                                                       1,
                                                                       Sum(_x) +
                                                                       1,
                                                                     ),
                                                              ), _x)), 0)

Examples

Basic #

the partition $4 + 2 + 1$ of $7$

live ↗

In$\mathrm{SmirnovWordCount}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$3$

the partition $3 + 3 + 1$ of $7$

live ↗

In$\mathrm{SmirnovWordCount}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$18$

Properties #

FindStat's value for [2]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([2]), \text{SmirnovWordCount})$Out$0$

Known value · FindStat St001097

FindStat's value for [3]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([3]), \text{SmirnovWordCount})$Out$0$

Known value · FindStat St001097

FindStat's value for [4]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([4]), \text{SmirnovWordCount})$Out$0$

Known value · FindStat St001097

About

Examples	5
Crosswalk	FindStat