- FindStat

Identifier

St000212: Integer partitions ⟶ ℤ

Values

=>
                            Cc0002;cc-rep
                            
                            
                            

                        []=>1
[1]=>1
[2]=>0
[1,1]=>1
[3]=>0
[2,1]=>1
[1,1,1]=>1
[4]=>0
[3,1]=>0
[2,2]=>1
[2,1,1]=>2
[1,1,1,1]=>1
[5]=>0
[4,1]=>0
[3,2]=>1
[3,1,1]=>1
[2,2,1]=>3
[2,1,1,1]=>3
[1,1,1,1,1]=>1
[6]=>0
[5,1]=>0
[4,2]=>0
[4,1,1]=>0
[3,3]=>1
[3,2,1]=>5
[3,1,1,1]=>3
[2,2,2]=>2
[2,2,1,1]=>6
[2,1,1,1,1]=>4
[1,1,1,1,1,1]=>1
[7]=>0
[6,1]=>0
[5,2]=>0
[5,1,1]=>0
[4,3]=>1
[4,2,1]=>2
[4,1,1,1]=>1
[3,3,1]=>5
[3,2,2]=>6
[3,2,1,1]=>14
[3,1,1,1,1]=>6
[2,2,2,1]=>8
[2,2,1,1,1]=>10
[2,1,1,1,1,1]=>5
[1,1,1,1,1,1,1]=>1
[8]=>0
[7,1]=>0
[6,2]=>0
[6,1,1]=>0
[5,3]=>0
[5,2,1]=>0
[5,1,1,1]=>0
[4,4]=>1
[4,3,1]=>8
[4,2,2]=>6
[4,2,1,1]=>11
[4,1,1,1,1]=>4
[3,3,2]=>10
[3,3,1,1]=>16
[3,2,2,1]=>28
[3,2,1,1,1]=>30
[3,1,1,1,1,1]=>10
[2,2,2,2]=>6
[2,2,2,1,1]=>18
[2,2,1,1,1,1]=>15
[2,1,1,1,1,1,1]=>6
[1,1,1,1,1,1,1,1]=>1
[9]=>0
[8,1]=>0
[7,2]=>0
[7,1,1]=>0
[6,3]=>0
[6,2,1]=>0
[6,1,1,1]=>0
[5,4]=>1
[5,3,1]=>3
[5,2,2]=>2
[5,2,1,1]=>3
[5,1,1,1,1]=>1
[4,4,1]=>7
[4,3,2]=>23
[4,3,1,1]=>34
[4,2,2,1]=>37
[4,2,1,1,1]=>35
[4,1,1,1,1,1]=>10
[3,3,3]=>6
[3,3,2,1]=>54
[3,3,1,1,1]=>40
[3,2,2,2]=>28
[3,2,2,1,1]=>76
[3,2,1,1,1,1]=>55
[3,1,1,1,1,1,1]=>15
[2,2,2,2,1]=>24
[2,2,2,1,1,1]=>33
[2,2,1,1,1,1,1]=>21
[2,1,1,1,1,1,1,1]=>7
[1,1,1,1,1,1,1,1,1]=>1
[10]=>0
[9,1]=>0
[8,2]=>0
[8,1,1]=>0
[7,3]=>0
[7,2,1]=>0
[7,1,1,1]=>0
[6,4]=>0
[6,3,1]=>0
[6,2,2]=>0
[6,2,1,1]=>0
[6,1,1,1,1]=>0
[5,5]=>1
[5,4,1]=>11
[5,3,2]=>18
[5,3,1,1]=>24
[5,2,2,1]=>22
[5,2,1,1,1]=>19
[5,1,1,1,1,1]=>5
[4,4,2]=>23
[4,4,1,1]=>31
[4,3,3]=>22
[4,3,2,1]=>148
[4,3,1,1,1]=>105
[4,2,2,2]=>53
[4,2,2,1,1]=>130
[4,2,1,1,1,1]=>85
[4,1,1,1,1,1,1]=>20
[3,3,3,1]=>44
[3,3,2,2]=>72
[3,3,2,1,1]=>170
[3,3,1,1,1,1]=>85
[3,2,2,2,1]=>128
[3,2,2,1,1,1]=>164
[3,2,1,1,1,1,1]=>91
[3,1,1,1,1,1,1,1]=>21
[2,2,2,2,2]=>18
[2,2,2,2,1,1]=>57
[2,2,2,1,1,1,1]=>54
[2,2,1,1,1,1,1,1]=>28
[2,1,1,1,1,1,1,1,1]=>8
[1,1,1,1,1,1,1,1,1,1]=>1
[11]=>0
[10,1]=>0
[9,2]=>0
[9,1,1]=>0
[8,3]=>0
[8,2,1]=>0
[8,1,1,1]=>0
[7,4]=>0
[7,3,1]=>0
[7,2,2]=>0
[7,2,1,1]=>0
[7,1,1,1,1]=>0
[6,5]=>1
[6,4,1]=>4
[6,3,2]=>5
[6,3,1,1]=>6
[6,2,2,1]=>5
[6,2,1,1,1]=>4
[6,1,1,1,1,1]=>1
[5,5,1]=>9
[5,4,2]=>49
[5,4,1,1]=>63
[5,3,3]=>31
[5,3,2,1]=>158
[5,3,1,1,1]=>107
[5,2,2,2]=>51
[5,2,2,1,1]=>113
[5,2,1,1,1,1]=>69
[5,1,1,1,1,1,1]=>15
[4,4,3]=>39
[4,4,2,1]=>165
[4,4,1,1,1]=>105
[4,3,3,1]=>180
[4,3,2,2]=>250
[4,3,2,1,1]=>553
[4,3,1,1,1,1]=>265
[4,2,2,2,1]=>287
[4,2,2,1,1,1]=>346
[4,2,1,1,1,1,1]=>175
[4,1,1,1,1,1,1,1]=>35
[3,3,3,2]=>110
[3,3,3,1,1]=>174
[3,3,2,2,1]=>370
[3,3,2,1,1,1]=>419
[3,3,1,1,1,1,1]=>161
[3,2,2,2,2]=>118
[3,2,2,2,1,1]=>349
[3,2,2,1,1,1,1]=>309
[3,2,1,1,1,1,1,1]=>140
[3,1,1,1,1,1,1,1,1]=>28
[2,2,2,2,2,1]=>75
[2,2,2,2,1,1,1]=>111
[2,2,2,1,1,1,1,1]=>82
[2,2,1,1,1,1,1,1,1]=>36
[2,1,1,1,1,1,1,1,1,1]=>9
[1,1,1,1,1,1,1,1,1,1,1]=>1
[12]=>0
[11,1]=>0
[10,2]=>0
[10,1,1]=>0
[9,3]=>0
[9,2,1]=>0
[9,1,1,1]=>0
[8,4]=>0
[8,3,1]=>0
[8,2,2]=>0
[8,2,1,1]=>0
[8,1,1,1,1]=>0
[7,5]=>0
[7,4,1]=>0
[7,3,2]=>0
[7,3,1,1]=>0
[7,2,2,1]=>0
[7,2,1,1,1]=>0
[7,1,1,1,1,1]=>0
[6,6]=>1
[6,5,1]=>14
[6,4,2]=>35
[6,4,1,1]=>42
[6,3,3]=>20
[6,3,2,1]=>80
[6,3,1,1,1]=>52
[6,2,2,2]=>25
[6,2,2,1,1]=>50
[6,2,1,1,1,1]=>29
[6,1,1,1,1,1,1]=>6
[5,5,2]=>41
[5,5,1,1]=>51
[5,4,3]=>109
[5,4,2,1]=>413
[5,4,1,1,1]=>257
[5,3,3,1]=>301
[5,3,2,2]=>369
[5,3,2,1,1]=>761
[5,3,1,1,1,1]=>351
[5,2,2,2,1]=>347
[5,2,2,1,1,1]=>397
[5,2,1,1,1,1,1]=>189
[5,1,1,1,1,1,1,1]=>35
[4,4,4]=>22
[4,4,3,1]=>353
[4,4,2,2]=>339
[4,4,2,1,1]=>693
[4,4,1,1,1,1]=>295
[4,3,3,2]=>518
[4,3,3,1,1]=>802
[4,3,2,2,1]=>1460
[4,3,2,1,1,1]=>1583
[4,3,1,1,1,1,1]=>581
[4,2,2,2,2]=>334
[4,2,2,2,1,1]=>925
[4,2,2,1,1,1,1]=>776
[4,2,1,1,1,1,1,1]=>322
[4,1,1,1,1,1,1,1,1]=>56
[3,3,3,3]=>72
[3,3,3,2,1]=>654
[3,3,3,1,1,1]=>508
[3,3,2,2,2]=>416
[3,3,2,2,1,1]=>1138
[3,3,2,1,1,1,1]=>889
[3,3,1,1,1,1,1,1]=>280
[3,2,2,2,2,1]=>542
[3,2,2,2,1,1,1]=>769
[3,2,2,1,1,1,1,1]=>531
[3,2,1,1,1,1,1,1,1]=>204
[3,1,1,1,1,1,1,1,1,1]=>36
[2,2,2,2,2,2]=>57
[2,2,2,2,2,1,1]=>186
[2,2,2,2,1,1,1,1]=>193
[2,2,2,1,1,1,1,1,1]=>118
[2,2,1,1,1,1,1,1,1,1]=>45
[2,1,1,1,1,1,1,1,1,1,1]=>10
[1,1,1,1,1,1,1,1,1,1,1,1]=>1
                    

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Description

The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row.
Summing over all partitions of $n$ yields the sequence
$$1, 1, 1, 2, 4, 9, 22, 59, 170, 516, 1658, \dots$$
which is oeis:A237770.
The references in this sequence of the OEIS indicate a connection with Baxter permutations.

References

[1] Dulucq, S., Guibert, O. Stack words, standard tableaux and Baxter permutations MathSciNet:1417289 DOI:10.1016/s0012-365x(96)83009-3

Code

def statistic(x):
    l = [any(any(f == e+1 for (e,f) in zip(row, row[1:])) 
             for row in T) 
         for T in StandardTableaux(x)]
    return l.count(False)

Created

Aug 21, 2014 at 14:22 by Per Alexandersson

Updated

Oct 29, 2017 at 16:35 by Martin Rubey