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Identifier

St000277: Integer compositions ⟶ ℤ

Values

[1] => 1

[1,1] => 1

[2] => 1

[1,1,1] => 1

[1,2] => 2

[2,1] => 2

[3] => 1

[1,1,1,1] => 1

[1,1,2] => 3

[1,2,1] => 5

[1,3] => 3

[2,1,1] => 3

[2,2] => 5

[3,1] => 3

[4] => 1

[1,1,1,1,1] => 1

[1,1,1,2] => 4

[1,1,2,1] => 9

[1,1,3] => 6

[1,2,1,1] => 9

[1,2,2] => 16

[1,3,1] => 11

[1,4] => 4

[2,1,1,1] => 4

[2,1,2] => 11

[2,2,1] => 16

[2,3] => 9

[3,1,1] => 6

[3,2] => 9

[4,1] => 4

[5] => 1

[1,1,1,1,1,1] => 1

[1,1,1,1,2] => 5

[1,1,1,2,1] => 14

[1,1,1,3] => 10

[1,1,2,1,1] => 19

[1,1,2,2] => 35

[1,1,3,1] => 26

[1,1,4] => 10

[1,2,1,1,1] => 14

[1,2,1,2] => 40

[1,2,2,1] => 61

[1,2,3] => 35

[1,3,1,1] => 26

[1,3,2] => 40

[1,4,1] => 19

[1,5] => 5

[2,1,1,1,1] => 5

[2,1,1,2] => 19

[2,1,2,1] => 40

[2,1,3] => 26

[2,2,1,1] => 35

[2,2,2] => 61

[2,3,1] => 40

[2,4] => 14

[3,1,1,1] => 10

[3,1,2] => 26

[3,2,1] => 35

[3,3] => 19

[4,1,1] => 10

[4,2] => 14

[5,1] => 5

[6] => 1

[1,1,1,1,1,1,1] => 1

[1,1,1,1,1,2] => 6

[1,1,1,1,2,1] => 20

[1,1,1,1,3] => 15

[1,1,1,2,1,1] => 34

[1,1,1,2,2] => 64

[1,1,1,3,1] => 50

[1,1,1,4] => 20

[1,1,2,1,1,1] => 34

[1,1,2,1,2] => 99

[1,1,2,2,1] => 155

[1,1,2,3] => 90

[1,1,3,1,1] => 71

[1,1,3,2] => 111

[1,1,4,1] => 55

[1,1,5] => 15

[1,2,1,1,1,1] => 20

[1,2,1,1,2] => 78

[1,2,1,2,1] => 169

[1,2,1,3] => 111

[1,2,2,1,1] => 155

[1,2,2,2] => 272

[1,2,3,1] => 181

[1,2,4] => 64

[1,3,1,1,1] => 50

[1,3,1,2] => 132

[1,3,2,1] => 181

[1,3,3] => 99

[1,4,1,1] => 55

[1,4,2] => 78

[1,5,1] => 29

[1,6] => 6

[2,1,1,1,1,1] => 6

[2,1,1,1,2] => 29

[2,1,1,2,1] => 78

[2,1,1,3] => 55

[2,1,2,1,1] => 99

[2,1,2,2] => 181

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Description

The number of ribbon shaped standard tableaux.
A ribbon is a connected skew shape which does not contain a $2\times 2$ square. The set of ribbon shapes are therefore in bijection with integer compositons, the parts of the composition specify the row lengths. This statistic records the number of standard tableaux of the given shape.
This is also the size of the preimage of the map 'descent composition' Mp00071descent composition from permutations to integer compositions: reading a tableau from bottom to top we obtain a permutation whose descent set is as prescribed.
For a composition $c=c_1,\dots,c_k$ of $n$, the number of ribbon shaped standard tableaux equals
$$ \sum_d (-1)^{k-\ell} \binom{n}{d_1, d_2, \dots, d_\ell}, $$
where the sum is over all coarsenings of $c$ obtained by replacing consecutive summands by their sum, see [sec 14.4, 1]

References

[1] Stanley, R. P. Enumerative combinatorics. Volume 1 MathSciNet:2868112

Code

def composition_to_ribbon(c):
    inner = []
    outer = []
    indent = 0
    for p in reversed(c):
        if indent > 0:
            inner.append(indent)
        outer.append(p+indent)
        indent += p-1
    return SkewPartition([outer[::-1], inner[::-1]])

def statistic(c):
    return StandardSkewTableaux(composition_to_ribbon(c)).cardinality()

# alternative implementation

def descents_composition(elt):
    if len(elt) == 0:
        return Composition([])
    d = [-1] + elt.descents() + [len(elt)-1]
    return Composition([ d[i+1]-d[i] for i in range(len(d)-1)])

@cached_function
def preimages(level):
    result = dict()
    for el in Permutations(level):
        image = descents_composition(el)
        result[image] = result.get(image, 0) + 1
    return result

def statistic(x):
    return preimages(x.size()).get(x, 0)

Created

Sep 11, 2015 at 21:12 by Martin Rubey

Updated

May 20, 2021 at 17:25 by Martin Rubey