- FindStat

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
[2,1,3,1]=>132
[2,1,4]=>50
[2,2,1,1,1]=>64
[2,2,1,2]=>181
[2,2,2,1]=>272
[2,2,3]=>155
[2,3,1,1]=>111
[2,3,2]=>169
[2,4,1]=>78
[2,5]=>20
[3,1,1,1,1]=>15
[3,1,1,2]=>55
[3,1,2,1]=>111
[3,1,3]=>71
[3,2,1,1]=>90
[3,2,2]=>155
[3,3,1]=>99
[3,4]=>34
[4,1,1,1]=>20
[4,1,2]=>50
[4,2,1]=>64
[4,3]=>34
[5,1,1]=>15
[5,2]=>20
[6,1]=>6
[7]=>1
[1,1,1,1,1,1,1,1]=>1
[1,1,1,1,1,1,2]=>7
[1,1,1,1,1,2,1]=>27
[1,1,1,1,1,3]=>21
[1,1,1,1,2,1,1]=>55
[1,1,1,1,2,2]=>105
[1,1,1,1,3,1]=>85
[1,1,1,1,4]=>35
[1,1,1,2,1,1,1]=>69
[1,1,1,2,1,2]=>203
[1,1,1,2,2,1]=>323
[1,1,1,2,3]=>189
[1,1,1,3,1,1]=>155
[1,1,1,3,2]=>245
[1,1,1,4,1]=>125
[1,1,1,5]=>35
[1,1,2,1,1,1,1]=>55
[1,1,2,1,1,2]=>217
[1,1,2,1,2,1]=>477
[1,1,2,1,3]=>315
[1,1,2,2,1,1]=>449
[1,1,2,2,2]=>791
[1,1,2,3,1]=>531
[1,1,2,4]=>189
[1,1,3,1,1,1]=>155
[1,1,3,1,2]=>413
[1,1,3,2,1]=>573
[1,1,3,3]=>315
[1,1,4,1,1]=>181
[1,1,4,2]=>259
[1,1,5,1]=>99
[1,1,6]=>21
[1,2,1,1,1,1,1]=>27
[1,2,1,1,1,2]=>133
[1,2,1,1,2,1]=>365
[1,2,1,1,3]=>259
[1,2,1,2,1,1]=>477
[1,2,1,2,2]=>875
[1,2,1,3,1]=>643
[1,2,1,4]=>245
[1,2,2,1,1,1]=>323
[1,2,2,1,2]=>917
[1,2,2,2,1]=>1385
[1,2,2,3]=>791
[1,2,3,1,1]=>573
[1,2,3,2]=>875
[1,2,4,1]=>407
[1,2,5]=>105
[1,3,1,1,1,1]=>85
[1,3,1,1,2]=>315
[1,3,1,2,1]=>643
[1,3,1,3]=>413
[1,3,2,1,1]=>531
[1,3,2,2]=>917
[1,3,3,1]=>589
[1,3,4]=>203
[1,4,1,1,1]=>125
[1,4,1,2]=>315
[1,4,2,1]=>407
[1,4,3]=>217
[1,5,1,1]=>99
[1,5,2]=>133
[1,6,1]=>41
[1,7]=>7
[2,1,1,1,1,1,1]=>7
[2,1,1,1,1,2]=>41
[2,1,1,1,2,1]=>133
[2,1,1,1,3]=>99
[2,1,1,2,1,1]=>217
[2,1,1,2,2]=>407
[2,1,1,3,1]=>315
[2,1,1,4]=>125
[2,1,2,1,1,1]=>203
[2,1,2,1,2]=>589
[2,1,2,2,1]=>917
[2,1,2,3]=>531
[2,1,3,1,1]=>413
[2,1,3,2]=>643
[2,1,4,1]=>315
[2,1,5]=>85
[2,2,1,1,1,1]=>105
[2,2,1,1,2]=>407
[2,2,1,2,1]=>875
[2,2,1,3]=>573
[2,2,2,1,1]=>791
[2,2,2,2]=>1385
[2,2,3,1]=>917
[2,2,4]=>323
[2,3,1,1,1]=>245
[2,3,1,2]=>643
[2,3,2,1]=>875
[2,3,3]=>477
[2,4,1,1]=>259
[2,4,2]=>365
[2,5,1]=>133
[2,6]=>27
[3,1,1,1,1,1]=>21
[3,1,1,1,2]=>99
[3,1,1,2,1]=>259
[3,1,1,3]=>181
[3,1,2,1,1]=>315
[3,1,2,2]=>573
[3,1,3,1]=>413
[3,1,4]=>155
[3,2,1,1,1]=>189
[3,2,1,2]=>531
[3,2,2,1]=>791
[3,2,3]=>449
[3,3,1,1]=>315
[3,3,2]=>477
[3,4,1]=>217
[3,5]=>55
[4,1,1,1,1]=>35
[4,1,1,2]=>125
[4,1,2,1]=>245
[4,1,3]=>155
[4,2,1,1]=>189
[4,2,2]=>323
[4,3,1]=>203
[4,4]=>69
[5,1,1,1]=>35
[5,1,2]=>85
[5,2,1]=>105
[5,3]=>55
[6,1,1]=>21
[6,2]=>27
[7,1]=>7
[8]=>1
[1,1,1,1,1,1,1,1,1]=>1
[1,1,1,1,1,1,1,2]=>8
[1,1,1,1,1,1,2,1]=>35
[1,1,1,1,1,1,3]=>28
[1,1,1,1,1,2,1,1]=>83
[1,1,1,1,1,2,2]=>160
[1,1,1,1,1,3,1]=>133
[1,1,1,1,1,4]=>56
[1,1,1,1,2,1,1,1]=>125
[1,1,1,1,2,1,2]=>370
[1,1,1,1,2,2,1]=>595
[1,1,1,1,2,3]=>350
[1,1,1,1,3,1,1]=>295
[1,1,1,1,3,2]=>470
[1,1,1,1,4,1]=>245
[1,1,1,1,5]=>70
[1,1,1,2,1,1,1,1]=>125
[1,1,1,2,1,1,2]=>496
[1,1,1,2,1,2,1]=>1099
[1,1,1,2,1,3]=>728
[1,1,1,2,2,1,1]=>1051
[1,1,1,2,2,2]=>1856
[1,1,1,2,3,1]=>1253
[1,1,1,2,4]=>448
[1,1,1,3,1,1,1]=>379
[1,1,1,3,1,2]=>1016
[1,1,1,3,2,1]=>1421
[1,1,1,3,3]=>784
[1,1,1,4,1,1]=>461
[1,1,1,4,2]=>664
[1,1,1,5,1]=>259
[1,1,1,6]=>56
[1,1,2,1,1,1,1,1]=>83
[1,1,2,1,1,1,2]=>412
[1,1,2,1,1,2,1]=>1141
[1,1,2,1,1,3]=>812
[1,1,2,1,2,1,1]=>1513
[1,1,2,1,2,2]=>2780
[1,1,2,1,3,1]=>2051
[1,1,2,1,4]=>784
[1,1,2,2,1,1,1]=>1051
[1,1,2,2,1,2]=>2990
[1,1,2,2,2,1]=>4529
[1,1,2,2,3]=>2590
[1,1,2,3,1,1]=>1889
[1,1,2,3,2]=>2890
[1,1,2,4,1]=>1351
[1,1,2,5]=>350
[1,1,3,1,1,1,1]=>295
[1,1,3,1,1,2]=>1100
[1,1,3,1,2,1]=>2261
[1,1,3,1,3]=>1456
[1,1,3,2,1,1]=>1889
[1,1,3,2,2]=>3268
[1,1,3,3,1]=>2107
[1,1,3,4]=>728
[1,1,4,1,1,1]=>461
[1,1,4,1,2]=>1168
[1,1,4,2,1]=>1519
[1,1,4,3]=>812
[1,1,5,1,1]=>379
[1,1,5,2]=>512
[1,1,6,1]=>161
[1,1,7]=>28
[1,2,1,1,1,1,1,1]=>35
[1,2,1,1,1,1,2]=>208
[1,2,1,1,1,2,1]=>685
[1,2,1,1,1,3]=>512
[1,2,1,1,2,1,1]=>1141
[1,2,1,1,2,2]=>2144
[1,2,1,1,3,1]=>1667
[1,2,1,1,4]=>664
[1,2,1,2,1,1,1]=>1099
[1,2,1,2,1,2]=>3194
[1,2,1,2,2,1]=>4985
[1,2,1,2,3]=>2890
[1,2,1,3,1,1]=>2261
[1,2,1,3,2]=>3526
[1,2,1,4,1]=>1735
[1,2,1,5]=>470
[1,2,2,1,1,1,1]=>595
[1,2,2,1,1,2]=>2312
[1,2,2,1,2,1]=>4985
[1,2,2,1,3]=>3268
[1,2,2,2,1,1]=>4529
[1,2,2,2,2]=>7936
[1,2,2,3,1]=>5263
[1,2,2,4]=>1856
[1,2,3,1,1,1]=>1421
[1,2,3,1,2]=>3736
[1,2,3,2,1]=>5095
[1,2,3,3]=>2780
[1,2,4,1,1]=>1519
[1,2,4,2]=>2144
[1,2,5,1]=>785
[1,2,6]=>160
[1,3,1,1,1,1,1]=>133
[1,3,1,1,1,2]=>632
[1,3,1,1,2,1]=>1667
[1,3,1,1,3]=>1168
[1,3,1,2,1,1]=>2051
[1,3,1,2,2]=>3736
[1,3,1,3,1]=>2701
[1,3,1,4]=>1016
[1,3,2,1,1,1]=>1253
[1,3,2,1,2]=>3526
[1,3,2,2,1]=>5263
[1,3,2,3]=>2990
[1,3,3,1,1]=>2107
[1,3,3,2]=>3194
[1,3,4,1]=>1457
[1,3,5]=>370
[1,4,1,1,1,1]=>245
[1,4,1,1,2]=>880
[1,4,1,2,1]=>1735
[1,4,1,3]=>1100
[1,4,2,1,1]=>1351
[1,4,2,2]=>2312
[1,4,3,1]=>1457
[1,4,4]=>496
[1,5,1,1,1]=>259
[1,5,1,2]=>632
[1,5,2,1]=>785
[1,5,3]=>412
[1,6,1,1]=>161
[1,6,2]=>208
[1,7,1]=>55
[1,8]=>8
[2,1,1,1,1,1,1,1]=>8
[2,1,1,1,1,1,2]=>55
[2,1,1,1,1,2,1]=>208
[2,1,1,1,1,3]=>161
[2,1,1,1,2,1,1]=>412
[2,1,1,1,2,2]=>785
[2,1,1,1,3,1]=>632
[2,1,1,1,4]=>259
[2,1,1,2,1,1,1]=>496
[2,1,1,2,1,2]=>1457
[2,1,1,2,2,1]=>2312
[2,1,1,2,3]=>1351
[2,1,1,3,1,1]=>1100
[2,1,1,3,2]=>1735
[2,1,1,4,1]=>880
[2,1,1,5]=>245
[2,1,2,1,1,1,1]=>370
[2,1,2,1,1,2]=>1457
[2,1,2,1,2,1]=>3194
[2,1,2,1,3]=>2107
[2,1,2,2,1,1]=>2990
[2,1,2,2,2]=>5263
[2,1,2,3,1]=>3526
[2,1,2,4]=>1253
[2,1,3,1,1,1]=>1016
[2,1,3,1,2]=>2701
[2,1,3,2,1]=>3736
[2,1,3,3]=>2051
[2,1,4,1,1]=>1168
[2,1,4,2]=>1667
[2,1,5,1]=>632
[2,1,6]=>133
[2,2,1,1,1,1,1]=>160
[2,2,1,1,1,2]=>785
[2,2,1,1,2,1]=>2144
[2,2,1,1,3]=>1519
[2,2,1,2,1,1]=>2780
[2,2,1,2,2]=>5095
[2,2,1,3,1]=>3736
[2,2,1,4]=>1421
[2,2,2,1,1,1]=>1856
[2,2,2,1,2]=>5263
[2,2,2,2,1]=>7936
[2,2,2,3]=>4529
[2,2,3,1,1]=>3268
[2,2,3,2]=>4985
[2,2,4,1]=>2312
[2,2,5]=>595
[2,3,1,1,1,1]=>470
[2,3,1,1,2]=>1735
[2,3,1,2,1]=>3526
[2,3,1,3]=>2261
[2,3,2,1,1]=>2890
[2,3,2,2]=>4985
[2,3,3,1]=>3194
[2,3,4]=>1099
[2,4,1,1,1]=>664
[2,4,1,2]=>1667
[2,4,2,1]=>2144
[2,4,3]=>1141
[2,5,1,1]=>512
[2,5,2]=>685
[2,6,1]=>208
[2,7]=>35
[3,1,1,1,1,1,1]=>28
[3,1,1,1,1,2]=>161
[3,1,1,1,2,1]=>512
[3,1,1,1,3]=>379
[3,1,1,2,1,1]=>812
[3,1,1,2,2]=>1519
[3,1,1,3,1]=>1168
[3,1,1,4]=>461
[3,1,2,1,1,1]=>728
[3,1,2,1,2]=>2107
[3,1,2,2,1]=>3268
[3,1,2,3]=>1889
[3,1,3,1,1]=>1456
[3,1,3,2]=>2261
[3,1,4,1]=>1100
[3,1,5]=>295
[3,2,1,1,1,1]=>350
[3,2,1,1,2]=>1351
[3,2,1,2,1]=>2890
[3,2,1,3]=>1889
[3,2,2,1,1]=>2590
[3,2,2,2]=>4529
[3,2,3,1]=>2990
[3,2,4]=>1051
[3,3,1,1,1]=>784
[3,3,1,2]=>2051
[3,3,2,1]=>2780
[3,3,3]=>1513
[3,4,1,1]=>812
[3,4,2]=>1141
[3,5,1]=>412
[3,6]=>83
[4,1,1,1,1,1]=>56
[4,1,1,1,2]=>259
[4,1,1,2,1]=>664
[4,1,1,3]=>461
[4,1,2,1,1]=>784
[4,1,2,2]=>1421
[4,1,3,1]=>1016
[4,1,4]=>379
[4,2,1,1,1]=>448
[4,2,1,2]=>1253
[4,2,2,1]=>1856
[4,2,3]=>1051
[4,3,1,1]=>728
[4,3,2]=>1099
[4,4,1]=>496
[4,5]=>125
[5,1,1,1,1]=>70
[5,1,1,2]=>245
[5,1,2,1]=>470
[5,1,3]=>295
[5,2,1,1]=>350
[5,2,2]=>595
[5,3,1]=>370
[5,4]=>125
[6,1,1,1]=>56
[6,1,2]=>133
[6,2,1]=>160
[6,3]=>83
[7,1,1]=>28
[7,2]=>35
[8,1]=>8
[9]=>1
[1,1,1,1,1,1,1,1,1,1]=>1
[1,1,1,1,1,1,2,2]=>231
[1,1,1,1,2,1,1,2]=>999
[1,1,1,1,2,2,1,1]=>2149
[1,1,1,1,3,3]=>1674
[1,1,1,1,5,1]=>574
[1,1,2,1,1,1,1,2]=>711
[1,1,2,1,1,2,1,1]=>3961
[1,1,2,1,2,3]=>10206
[1,1,2,2,1,1,1,1]=>2149
[1,1,2,2,2,2]=>28839
[1,1,3,1,1,3]=>4536
[1,1,3,2,1,2]=>13941
[1,1,3,3,1,1]=>8371
[1,1,4,4]=>2064
[1,1,7,1]=>244
[1,2,1,1,4,1]=>5191
[1,2,6,1]=>1369
[1,3,1,1,3,1]=>8371
[1,3,5,1]=>3079
[1,4,1,1,2,1]=>5191
[1,4,4,1]=>3961
[1,5,1,1,1,1]=>574
[1,5,3,1]=>3079
[1,6,1,2]=>1134
[1,6,2,1]=>1369
[1,7,1,1]=>244
[1,8,1]=>71
[1,9]=>9
[2,1,1,1,1,1,1,2]=>71
[2,1,1,1,1,2,1,1]=>711
[2,1,1,1,2,3]=>2906
[2,1,1,2,1,1,1,1]=>999
[2,1,1,2,2,2]=>14759
[2,1,2,1,1,3]=>6056
[2,1,2,2,1,2]=>22121
[2,1,2,3,1,1]=>13941
[2,1,3,4]=>5264
[2,1,6,1]=>1134
[2,2,1,1,1,1,1,1]=>231
[2,2,1,1,2,2]=>13991
[2,2,2,1,1,2]=>14759
[2,2,2,2,1,1]=>28839
[2,2,3,3]=>17594
[3,1,1,1,1,3]=>701
[3,1,1,2,1,2]=>6056
[3,1,1,3,1,1]=>4536
[3,1,2,4]=>4949
[3,2,1,1,1,2]=>2906
[3,2,1,2,1,1]=>10206
[3,2,2,3]=>16451
[3,3,1,1,1,1]=>1674
[3,3,2,2]=>17594
[4,1,1,4]=>1301
[4,2,1,3]=>4949
[4,3,1,2]=>5264
[4,4,1,1]=>2064
[5,5]=>251
[8,1,1]=>36
[9,1]=>9
[1,10]=>10
[1,8,1,1]=>351
[1,7,2,1]=>2221
[1,6,3,1]=>5851
[1,5,4,1]=>9151
[1,4,5,1]=>9151
[1,3,6,1]=>5851
[1,2,7,1]=>2221
[1,1,8,1]=>351
[10,1]=>10
[1,1,1,1,1,1,1,1,1,1,1,1]=>1
[1,1,1,1,1,1,1,1,2,2]=>429
[1,1,1,1,1,1,2,2,1,1]=>6909
[1,1,1,1,1,1,2,1,1,2]=>3157
[1,1,1,1,1,1,3,3]=>5698
[1,1,1,1,2,2,1,1,1,1]=>15049
[1,1,1,1,2,2,2,2]=>203181
[1,1,1,1,2,1,1,2,1,1]=>26709
[1,1,1,1,2,1,1,1,1,2]=>4741
[1,1,1,1,2,1,2,3]=>69850
[1,1,1,1,3,3,1,1]=>65154
[1,1,1,1,3,2,1,2]=>107866
[1,1,1,1,3,1,1,3]=>34375
[1,1,1,1,4,4]=>16995
[1,1,2,2,1,1,1,1,1,1]=>6909
[1,1,2,2,1,1,2,2]=>421421
[1,1,2,2,2,2,1,1]=>880529
[1,1,2,2,2,1,1,2]=>450021
[1,1,2,2,3,3]=>538890
[1,1,2,1,1,2,1,1,1,1]=>26709
[1,1,2,1,1,2,2,2]=>396077
[1,1,2,1,1,1,1,2,1,1]=>18041
[1,1,2,1,1,1,1,1,1,2]=>1749
[1,1,2,1,1,1,2,3]=>75570
[1,1,2,1,2,3,1,1]=>392822
[1,1,2,1,2,2,1,2]=>622314
[1,1,2,1,2,1,1,3]=>169455
[1,1,2,1,3,4]=>150227
[1,1,3,3,1,1,1,1]=>65154
[1,1,3,3,2,2]=>686906
[1,1,3,2,1,2,1,1]=>392822
[1,1,3,2,1,1,1,2]=>111474
[1,1,3,2,2,3]=>635745
[1,1,3,1,1,3,1,1]=>166541
[1,1,3,1,1,2,1,2]=>221529
[1,1,3,1,1,1,1,3]=>25080
[1,1,3,1,2,4]=>185108
[1,1,4,4,1,1]=>90509
[1,1,4,3,1,2]=>230241
[1,1,4,2,1,3]=>215160
[1,1,4,1,1,4]=>55220
[1,1,5,5]=>11825
[2,2,1,1,1,1,1,1,1,1]=>429
[2,2,1,1,1,1,2,2]=>66989
[2,2,1,1,2,2,1,1]=>421421
[2,2,1,1,2,1,1,2]=>199341
[2,2,1,1,3,3]=>312102
[2,2,2,2,1,1,1,1]=>203181
[2,2,2,2,2,2]=>2702765
[2,2,2,1,1,2,1,1]=>396077
[2,2,2,1,1,1,1,2]=>72621
[2,2,2,1,2,3]=>996390
[2,2,3,3,1,1]=>686906
[2,2,3,2,1,2]=>1152546
[2,2,3,1,1,3]=>385395
[2,2,4,4]=>157475
[2,1,1,2,1,1,1,1,1,1]=>3157
[2,1,1,2,1,1,2,2]=>199341
[2,1,1,2,2,2,1,1]=>450021
[2,1,1,2,2,1,1,2]=>228205
[2,1,1,2,3,3]=>280774
[2,1,1,1,1,2,1,1,1,1]=>4741
[2,1,1,1,1,2,2,2]=>72621
[2,1,1,1,1,1,1,2,1,1]=>1749
[2,1,1,1,1,1,1,1,1,2]=>109
[2,1,1,1,1,1,2,3]=>9910
[2,1,1,1,2,3,1,1]=>111474
[2,1,1,1,2,2,1,2]=>174514
[2,1,1,1,2,1,1,3]=>45715
[2,1,1,1,3,4]=>46947
[2,1,2,3,1,1,1,1]=>107866
[2,1,2,3,2,2]=>1152546
[2,1,2,2,1,2,1,1]=>622314
[2,1,2,2,1,1,1,2]=>174514
[2,1,2,2,2,3]=>1022845
[2,1,2,1,1,3,1,1]=>221529
[2,1,2,1,1,2,1,2]=>291169
[2,1,2,1,1,1,1,3]=>30700
[2,1,2,1,2,4]=>262932
[2,1,3,4,1,1]=>230241
[2,1,3,3,1,2]=>578665
[2,1,3,2,1,3]=>527284
[2,1,3,1,1,4]=>123540
[2,1,4,5]=>39225
[3,3,1,1,1,1,1,1]=>5698
[3,3,1,1,2,2]=>312102
[3,3,2,2,1,1]=>538890
[3,3,2,1,1,2]=>280774
[3,3,3,3]=>315523
[3,2,1,2,1,1,1,1]=>69850
[3,2,1,2,2,2]=>996390
[3,2,1,1,1,2,1,1]=>75570
[3,2,1,1,1,1,1,2]=>9910
[3,2,1,1,2,3]=>251371
[3,2,2,3,1,1]=>635745
[3,2,2,2,1,2]=>1022845
[3,2,2,1,1,3]=>294106
[3,2,3,4]=>215346
[3,1,1,3,1,1,1,1]=>34375
[3,1,1,3,2,2]=>385395
[3,1,1,2,1,2,1,1]=>169455
[3,1,1,2,1,1,1,2]=>45715
[3,1,1,2,2,3]=>294106
[3,1,1,1,1,3,1,1]=>25080
[3,1,1,1,1,2,1,2]=>30700
[3,1,1,1,1,1,1,3]=>1891
[3,1,1,1,2,4]=>43251
[3,1,2,4,1,1]=>215160
[3,1,2,3,1,2]=>527284
[3,1,2,2,1,3]=>458083
[3,1,2,1,1,4]=>90771
[3,1,3,5]=>54966
[4,4,1,1,1,1]=>16995
[4,4,2,2]=>157475
[4,3,1,2,1,1]=>150227
[4,3,1,1,1,2]=>46947
[4,3,2,3]=>215346
[4,2,1,3,1,1]=>185108
[4,2,1,2,1,2]=>262932
[4,2,1,1,1,3]=>43251
[4,2,2,4]=>147443
[4,1,1,4,1,1]=>55220
[4,1,1,3,1,2]=>123540
[4,1,1,2,1,3]=>90771
[4,1,1,1,1,4]=>8051
[4,1,2,5]=>36926
[5,5,1,1]=>11825
[5,4,1,2]=>39225
[5,3,1,3]=>54966
[5,2,1,4]=>36926
[5,1,1,5]=>8051
[6,6]=>923
                    

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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