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Statistic identifier: St000506
-----------------------------------------------------------------------------
Collection: Integer partitions
-----------------------------------------------------------------------------
Description: The number of standard desarrangement tableaux of shape equal to the given partition.
A '''standard desarrangement tableau''' is a standard tableau whose first ascent is even. Here, an ascent of a standard tableau is an entry $i$ such that $i+1$ appears to the right or above $i$ in the tableau (with respect to English tableau notation).
This is also the nullity of the random-to-random operator (and the random-to-top) operator acting on the simple module of the symmetric group indexed by the given partition. See also:
* [[St000046]]: The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition
* [[St000500]]: Eigenvalues of the random-to-random operator acting on the regular representation.
-----------------------------------------------------------------------------
References:
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Code:
def tableau_ascents(t):
r"""
The (sorted list) of ascents of the standard tableau `t`.
An *ascent* of a standard tableau `t` is an entry `i`
such that `i+1` apears to the right or above `i` in `t`
(in English notation for tableaux).
"""
locations = {}
for (i, row) in enumerate(t):
for (j, entry) in enumerate(row):
locations[entry] = (i, j)
ascents = [t.size()]
for i in range(1, t.size()):
# ascent means i+1 appears to the right or above
x, _ = locations[i]
u, _ = locations[i+1]
if u <= x:
ascents.append(i)
return sorted(ascents)
def is_desarrangement_tableau(t):
r"""
Test whether a tableau is a desarrangement tableau.
A *desarrangement tableau* is a standard tableau
whose first ascent is even.
"""
return min(tableau_ascents(Tableau(t))) % 2 == 0
def statistic(la):
return len([t for t in StandardTableaux(la) if is_desarrangement_tableau(t)])
-----------------------------------------------------------------------------
Statistic values:
[1] => 0
[2] => 0
[1,1] => 1
[3] => 0
[2,1] => 1
[1,1,1] => 0
[4] => 0
[3,1] => 1
[2,2] => 1
[2,1,1] => 1
[1,1,1,1] => 1
[5] => 0
[4,1] => 1
[3,2] => 2
[3,1,1] => 2
[2,2,1] => 2
[2,1,1,1] => 2
[1,1,1,1,1] => 0
[6] => 0
[5,1] => 1
[4,2] => 3
[4,1,1] => 3
[3,3] => 2
[3,2,1] => 6
[3,1,1,1] => 4
[2,2,2] => 2
[2,2,1,1] => 4
[2,1,1,1,1] => 2
[1,1,1,1,1,1] => 1
[7] => 0
[6,1] => 1
[5,2] => 4
[5,1,1] => 4
[4,3] => 5
[4,2,1] => 12
[4,1,1,1] => 7
[3,3,1] => 8
[3,2,2] => 8
[3,2,1,1] => 14
[3,1,1,1,1] => 6
[2,2,2,1] => 6
[2,2,1,1,1] => 6
[2,1,1,1,1,1] => 3
[1,1,1,1,1,1,1] => 0
[8] => 0
[7,1] => 1
[6,2] => 5
[6,1,1] => 5
[5,3] => 9
[5,2,1] => 20
[5,1,1,1] => 11
[4,4] => 5
[4,3,1] => 25
[4,2,2] => 20
[4,2,1,1] => 33
[4,1,1,1,1] => 13
[3,3,2] => 16
[3,3,1,1] => 22
[3,2,2,1] => 28
[3,2,1,1,1] => 26
[3,1,1,1,1,1] => 9
[2,2,2,2] => 6
[2,2,2,1,1] => 12
[2,2,1,1,1,1] => 9
[2,1,1,1,1,1,1] => 3
[1,1,1,1,1,1,1,1] => 1
[9] => 0
[8,1] => 1
[7,2] => 6
[7,1,1] => 6
[6,3] => 14
[6,2,1] => 30
[6,1,1,1] => 16
[5,4] => 14
[5,3,1] => 54
[5,2,2] => 40
[5,2,1,1] => 64
[5,1,1,1,1] => 24
[4,4,1] => 30
[4,3,2] => 61
[4,3,1,1] => 80
[4,2,2,1] => 81
[4,2,1,1,1] => 72
[4,1,1,1,1,1] => 22
[3,3,3] => 16
[3,3,2,1] => 66
[3,3,1,1,1] => 48
[3,2,2,2] => 34
[3,2,2,1,1] => 66
[3,2,1,1,1,1] => 44
[3,1,1,1,1,1,1] => 12
[2,2,2,2,1] => 18
[2,2,2,1,1,1] => 21
[2,2,1,1,1,1,1] => 12
[2,1,1,1,1,1,1,1] => 4
[1,1,1,1,1,1,1,1,1] => 0
[10] => 0
[9,1] => 1
[8,2] => 7
[8,1,1] => 7
[7,3] => 20
[7,2,1] => 42
[7,1,1,1] => 22
[6,4] => 28
[6,3,1] => 98
[6,2,2] => 70
[6,2,1,1] => 110
[6,1,1,1,1] => 40
[5,5] => 14
[5,4,1] => 98
[5,3,2] => 155
[5,3,1,1] => 198
[5,2,2,1] => 185
[5,2,1,1,1] => 160
[5,1,1,1,1,1] => 46
[4,4,2] => 91
[4,4,1,1] => 110
[4,3,3] => 77
[4,3,2,1] => 288
[4,3,1,1,1] => 200
[4,2,2,2] => 115
[4,2,2,1,1] => 219
[4,2,1,1,1,1] => 138
[4,1,1,1,1,1,1] => 34
[3,3,3,1] => 82
[3,3,2,2] => 100
[3,3,2,1,1] => 180
[3,3,1,1,1,1] => 92
[3,2,2,2,1] => 118
[3,2,2,1,1,1] => 131
[3,2,1,1,1,1,1] => 68
[3,1,1,1,1,1,1,1] => 16
[2,2,2,2,2] => 18
[2,2,2,2,1,1] => 39
[2,2,2,1,1,1,1] => 33
[2,2,1,1,1,1,1,1] => 16
[2,1,1,1,1,1,1,1,1] => 4
[1,1,1,1,1,1,1,1,1,1] => 1
[11] => 0
[10,1] => 1
[9,2] => 8
[9,1,1] => 8
[8,3] => 27
[8,2,1] => 56
[8,1,1,1] => 29
[7,4] => 48
[7,3,1] => 160
[7,2,2] => 112
[7,2,1,1] => 174
[7,1,1,1,1] => 62
[6,5] => 42
[6,4,1] => 224
[6,3,2] => 323
[6,3,1,1] => 406
[6,2,2,1] => 365
[6,2,1,1,1] => 310
[6,1,1,1,1,1] => 86
[5,5,1] => 112
[5,4,2] => 344
[5,4,1,1] => 406
[5,3,3] => 232
[5,3,2,1] => 826
[5,3,1,1,1] => 558
[5,2,2,2] => 300
[5,2,2,1,1] => 564
[5,2,1,1,1,1] => 344
[5,1,1,1,1,1,1] => 80
[4,4,3] => 168
[4,4,2,1] => 489
[4,4,1,1,1] => 310
[4,3,3,1] => 447
[4,3,2,2] => 503
[4,3,2,1,1] => 887
[4,3,1,1,1,1] => 430
[4,2,2,2,1] => 452
[4,2,2,1,1,1] => 488
[4,2,1,1,1,1,1] => 240
[4,1,1,1,1,1,1,1] => 50
[3,3,3,2] => 182
[3,3,3,1,1] => 262
[3,3,2,2,1] => 398
[3,3,2,1,1,1] => 403
[3,3,1,1,1,1,1] => 160
[3,2,2,2,2] => 136
[3,2,2,2,1,1] => 288
[3,2,2,1,1,1,1] => 232
[3,2,1,1,1,1,1,1] => 100
[3,1,1,1,1,1,1,1,1] => 20
[2,2,2,2,2,1] => 57
[2,2,2,2,1,1,1] => 72
[2,2,2,1,1,1,1,1] => 49
[2,2,1,1,1,1,1,1,1] => 20
[2,1,1,1,1,1,1,1,1,1] => 5
[1,1,1,1,1,1,1,1,1,1,1] => 0
[12] => 0
[11,1] => 1
[10,2] => 9
[10,1,1] => 9
[9,3] => 35
[9,2,1] => 72
[9,1,1,1] => 37
[8,4] => 75
[8,3,1] => 243
[8,2,2] => 168
[8,2,1,1] => 259
[8,1,1,1,1] => 91
[7,5] => 90
[7,4,1] => 432
[7,3,2] => 595
[7,3,1,1] => 740
[7,2,2,1] => 651
[7,2,1,1,1] => 546
[7,1,1,1,1,1] => 148
[6,6] => 42
[6,5,1] => 378
[6,4,2] => 891
[6,4,1,1] => 1036
[6,3,3] => 555
[6,3,2,1] => 1920
[6,3,1,1,1] => 1274
[6,2,2,2] => 665
[6,2,2,1,1] => 1239
[6,2,1,1,1,1] => 740
[6,1,1,1,1,1,1] => 166
[5,5,2] => 456
[5,5,1,1] => 518
[5,4,3] => 744
[5,4,2,1] => 2065
[5,4,1,1,1] => 1274
[5,3,3,1] => 1505
[5,3,2,2] => 1629
[5,3,2,1,1] => 2835
[5,3,1,1,1,1] => 1332
[5,2,2,2,1] => 1316
[5,2,2,1,1,1] => 1396
[5,2,1,1,1,1,1] => 664
[5,1,1,1,1,1,1,1] => 130
[4,4,4] => 168
[4,4,3,1] => 1104
[4,4,2,2] => 992
[4,4,2,1,1] => 1686
[4,4,1,1,1,1] => 740
[4,3,3,2] => 1132
[4,3,3,1,1] => 1596
[4,3,2,2,1] => 2240
[4,3,2,1,1,1] => 2208
[4,3,1,1,1,1,1] => 830
[4,2,2,2,2] => 588
[4,2,2,2,1,1] => 1228
[4,2,2,1,1,1,1] => 960
[4,2,1,1,1,1,1,1] => 390
[4,1,1,1,1,1,1,1,1] => 70
[3,3,3,3] => 182
[3,3,3,2,1] => 842
[3,3,3,1,1,1] => 665
[3,3,2,2,2] => 534
[3,3,2,2,1,1] => 1089
[3,3,2,1,1,1,1] => 795
[3,3,1,1,1,1,1,1] => 260
[3,2,2,2,2,1] => 481
[3,2,2,2,1,1,1] => 592
[3,2,2,1,1,1,1,1] => 381
[3,2,1,1,1,1,1,1,1] => 140
[3,1,1,1,1,1,1,1,1,1] => 25
[2,2,2,2,2,2] => 57
[2,2,2,2,2,1,1] => 129
[2,2,2,2,1,1,1,1] => 121
[2,2,2,1,1,1,1,1,1] => 69
[2,2,1,1,1,1,1,1,1,1] => 25
[2,1,1,1,1,1,1,1,1,1,1] => 5
[1,1,1,1,1,1,1,1,1,1,1,1] => 1
-----------------------------------------------------------------------------
Created: May 24, 2016 at 23:10 by Franco Saliola
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Last Updated: Jun 11, 2016 at 01:03 by Martin Rubey