OrderedRefinementCount

The partitions whose parts cut, in order, into consecutive blocks summing to the parts of this one.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000935via IntegerPartition✓↗

OrderedRefinementCount(partition)The partitions whose parts cut, in order, into consecutive blocks summing to the parts of this one.

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.

Definition

Epsil, as the implementation runs it.

on IntegerPartition, over _x

1 if Length(_x) < 1 else Apply(
                                (pre: list<integer>) => Apply(
                                                               (w: list<integer>) => Sum(w),
                                                               Fold(
                                                                     Function(Map(
                                                                                   Function(Map(
                                                                                                 Function(Sum(Map(Function(a[b - c +
                                                                                                                             1][m], m), Range(c, Sum(_x)))) if c <= b &&
                                                                                                                                                              Sum(Map(Function(1 if s >
                                                                                                                                                                                    b - c &&
                                                                                                                                                                                    s < b else 0, s), pre)) ==
                                                                                                                                                              0 else 0, c),
                                                                                                 Range(1, Sum(_x)),
                                                                                               ) if r ==
                                                                                                   b + 1 else a[r], r),
                                                                                   Range(1, Sum(_x) + 1),
                                                                                 ), a, b),
                                                                     Map(
                                                                          Function(Map(
                                                                                        Function(1 if r == 1 &&
                                                                                                      c == Sum(_x) else 0, c),
                                                                                        Range(1, Sum(_x)),
                                                                                      ), r),
                                                                          Range(
                                                                                 1,
                                                                                 Sum(_x) + 1,
                                                                               ),
                                                                        ),
                                                                     Range(
                                                                            1,
                                                                            Sum(_x),
                                                                          ),
                                                                   )[Sum(_x) + 1],
                                                             ),
                                Map(Function(Sum(Map(
                                                      Function(_x[q], q),
                                                      Range(1, i),
                                                    )), i), Range(1, Length(_x))),
                              )

Examples

Basic #

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

live ↗

In$\mathrm{OrderedRefinementCount}({\operatorname{IntegerPartition}}([4, 2, 1]))$Out$7$

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

live ↗

In$\mathrm{OrderedRefinementCount}({\operatorname{IntegerPartition}}([3, 3, 1]))$Out$5$

Properties #

FindStat's value for [1]

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{IntegerPartition}}([1]), \text{OrderedRefinementCount})$Out$1$

Known value · FindStat St000935

FindStat's value for [2]

live ↗

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

Known value · FindStat St000935

FindStat's value for [3]

live ↗

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

Known value · FindStat St000935

About

Examples	5
Crosswalk	FindStat