- FindStat

Identifier

St000275: Integer partitions ⟶ ℤ

Values

=>
                            Cc0002;cc-rep
                            
                            
                            

                        []=>1
[1]=>1
[2]=>1
[1,1]=>1
[3]=>1
[2,1]=>4
[1,1,1]=>1
[4]=>1
[3,1]=>7
[2,2]=>4
[2,1,1]=>11
[1,1,1,1]=>1
[5]=>1
[4,1]=>11
[3,2]=>15
[3,1,1]=>32
[2,2,1]=>34
[2,1,1,1]=>26
[1,1,1,1,1]=>1
[6]=>1
[5,1]=>16
[4,2]=>26
[4,1,1]=>76
[3,3]=>15
[3,2,1]=>192
[3,1,1,1]=>122
[2,2,2]=>34
[2,2,1,1]=>180
[2,1,1,1,1]=>57
[1,1,1,1,1,1]=>1
[7]=>1
[6,1]=>22
[5,2]=>42
[5,1,1]=>156
[4,3]=>56
[4,2,1]=>474
[4,1,1,1]=>426
[3,3,1]=>267
[3,2,2]=>294
[3,2,1,1]=>1494
[3,1,1,1,1]=>423
[2,2,2,1]=>496
[2,2,1,1,1]=>768
[2,1,1,1,1,1]=>120
[1,1,1,1,1,1,1]=>1
[8]=>1
[7,1]=>29
[6,2]=>64
[6,1,1]=>288
[5,3]=>98
[5,2,1]=>1038
[5,1,1,1]=>1206
[4,4]=>56
[4,3,1]=>1344
[4,2,2]=>768
[4,2,1,1]=>5142
[4,1,1,1,1]=>2127
[3,3,2]=>855
[3,3,1,1]=>2829
[3,2,2,1]=>5946
[3,2,1,1,1]=>9204
[3,1,1,1,1,1]=>1389
[2,2,2,2]=>496
[2,2,2,1,1]=>4288
[2,2,1,1,1,1]=>2904
[2,1,1,1,1,1,1]=>247
[1,1,1,1,1,1,1,1]=>1
[9]=>1
[8,1]=>37
[7,2]=>93
[7,1,1]=>491
[6,3]=>162
[6,2,1]=>2062
[6,1,1,1]=>2934
[5,4]=>210
[5,3,1]=>3068
[5,2,2]=>1806
[5,2,1,1]=>14988
[5,1,1,1,1]=>8157
[4,4,1]=>1736
[4,3,2]=>4590
[4,3,1,1]=>18864
[4,2,2,1]=>20838
[4,2,1,1,1]=>43422
[4,1,1,1,1,1]=>9897
[3,3,3]=>855
[3,3,2,1]=>22680
[3,3,1,1,1]=>23349
[3,2,2,2]=>7930
[3,2,2,1,1]=>70206
[3,2,1,1,1,1]=>49569
[3,1,1,1,1,1,1]=>4414
[2,2,2,2,1]=>11056
[2,2,2,1,1,1]=>28768
[2,2,1,1,1,1,1]=>10194
[2,1,1,1,1,1,1,1]=>502
[1,1,1,1,1,1,1,1,1]=>1
[10]=>1
[9,1]=>46
[8,2]=>130
[8,1,1]=>787
[7,3]=>255
[7,2,1]=>3788
[7,1,1,1]=>6371
[6,4]=>372
[6,3,1]=>6426
[6,2,2]=>3868
[6,2,1,1]=>38224
[6,1,1,1,1]=>25761
[5,5]=>210
[5,4,1]=>8220
[5,3,2]=>11270
[5,3,1,1]=>55328
[5,2,2,1]=>63456
[5,2,1,1,1]=>165978
[5,1,1,1,1,1]=>50682
[4,4,2]=>6326
[4,4,1,1]=>31016
[4,3,3]=>7155
[4,3,2,1]=>156894
[4,3,1,1,1]=>203304
[4,2,2,2]=>28768
[4,2,2,1,1]=>325500
[4,2,1,1,1,1]=>316164
[4,1,1,1,1,1,1]=>44002
[3,3,3,1]=>28665
[3,3,2,2]=>46470
[3,3,2,1,1]=>346539
[3,3,1,1,1,1]=>166314
[3,2,2,2,1]=>232216
[3,2,2,1,1,1]=>635610
[3,2,1,1,1,1,1]=>245148
[3,1,1,1,1,1,1,1]=>13744
[2,2,2,2,2]=>11056
[2,2,2,2,1,1]=>141584
[2,2,2,1,1,1,1]=>166042
[2,2,1,1,1,1,1,1]=>34096
[2,1,1,1,1,1,1,1,1]=>1013
[1,1,1,1,1,1,1,1,1,1]=>1
                    

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Description

Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition.

References

[1] Hivert, F., Novelli, J.-C., Thibon, J.-Y. Multivariate generalizations of the Foata-Schützenberger equidistribution MathSciNet:2509639 arXiv:math/0605060

Code

import collections
def part_of_perm(p):
    c = p.to_lehmer_code()
    return Partition(sorted([c.count(i) for i in range(len(p)) if i in c])[::-1])

@cached_function
def stat(N):
    res = collections.defaultdict(int)
    for p in Permutations(N):
        res[part_of_perm(p)] += 1
    return dict(res)

def statistic(L):
    return stat(L.size())[L]

Created

Sep 04, 2015 at 17:58 by Florent Hivert

Updated

Sep 15, 2015 at 15:49 by Christian Stump