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-----------------------------------------------------------------------------
Statistic identifier: St000212
-----------------------------------------------------------------------------
Collection: Integer partitions
-----------------------------------------------------------------------------
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)
-----------------------------------------------------------------------------
Statistic values:
[] => 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
-----------------------------------------------------------------------------
Created: Aug 21, 2014 at 14:22 by Per Alexandersson
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Last Updated: Oct 29, 2017 at 16:35 by Martin Rubey