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Your data matches 34 different statistics following compositions of up to 3 maps.
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Matching statistic: St000389
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(load all 3 compositions to match this statistic)
St000389: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0
1 => 1
00 => 0
01 => 1
10 => 1
11 => 0
000 => 0
001 => 1
010 => 1
011 => 0
100 => 1
101 => 2
110 => 0
111 => 1
0000 => 0
0001 => 1
0010 => 1
0011 => 0
0100 => 1
0101 => 2
0110 => 0
0111 => 1
1000 => 1
1001 => 2
1010 => 2
1011 => 1
1100 => 0
1101 => 1
1110 => 1
1111 => 0
00000 => 0
00001 => 1
00010 => 1
00011 => 0
00100 => 1
00101 => 2
00110 => 0
00111 => 1
01000 => 1
01001 => 2
01010 => 2
01011 => 1
01100 => 0
01101 => 1
01110 => 1
01111 => 0
10000 => 1
10001 => 2
10010 => 2
10011 => 1
Description
The number of runs of ones of odd length in a binary word.
Matching statistic: St000142
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [2]
=> 1
1 => [1,1] => [1,1]
=> 0
00 => [3] => [3]
=> 0
01 => [2,1] => [2,1]
=> 1
10 => [1,2] => [2,1]
=> 1
11 => [1,1,1] => [1,1,1]
=> 0
000 => [4] => [4]
=> 1
001 => [3,1] => [3,1]
=> 0
010 => [2,2] => [2,2]
=> 2
011 => [2,1,1] => [2,1,1]
=> 1
100 => [1,3] => [3,1]
=> 0
101 => [1,2,1] => [2,1,1]
=> 1
110 => [1,1,2] => [2,1,1]
=> 1
111 => [1,1,1,1] => [1,1,1,1]
=> 0
0000 => [5] => [5]
=> 0
0001 => [4,1] => [4,1]
=> 1
0010 => [3,2] => [3,2]
=> 1
0011 => [3,1,1] => [3,1,1]
=> 0
0100 => [2,3] => [3,2]
=> 1
0101 => [2,2,1] => [2,2,1]
=> 2
0110 => [2,1,2] => [2,2,1]
=> 2
0111 => [2,1,1,1] => [2,1,1,1]
=> 1
1000 => [1,4] => [4,1]
=> 1
1001 => [1,3,1] => [3,1,1]
=> 0
1010 => [1,2,2] => [2,2,1]
=> 2
1011 => [1,2,1,1] => [2,1,1,1]
=> 1
1100 => [1,1,3] => [3,1,1]
=> 0
1101 => [1,1,2,1] => [2,1,1,1]
=> 1
1110 => [1,1,1,2] => [2,1,1,1]
=> 1
1111 => [1,1,1,1,1] => [1,1,1,1,1]
=> 0
00000 => [6] => [6]
=> 1
00001 => [5,1] => [5,1]
=> 0
00010 => [4,2] => [4,2]
=> 2
00011 => [4,1,1] => [4,1,1]
=> 1
00100 => [3,3] => [3,3]
=> 0
00101 => [3,2,1] => [3,2,1]
=> 1
00110 => [3,1,2] => [3,2,1]
=> 1
00111 => [3,1,1,1] => [3,1,1,1]
=> 0
01000 => [2,4] => [4,2]
=> 2
01001 => [2,3,1] => [3,2,1]
=> 1
01010 => [2,2,2] => [2,2,2]
=> 3
01011 => [2,2,1,1] => [2,2,1,1]
=> 2
01100 => [2,1,3] => [3,2,1]
=> 1
01101 => [2,1,2,1] => [2,2,1,1]
=> 2
01110 => [2,1,1,2] => [2,2,1,1]
=> 2
01111 => [2,1,1,1,1] => [2,1,1,1,1]
=> 1
10000 => [1,5] => [5,1]
=> 0
10001 => [1,4,1] => [4,1,1]
=> 1
10010 => [1,3,2] => [3,2,1]
=> 1
10011 => [1,3,1,1] => [3,1,1,1]
=> 0
Description
The number of even parts of a partition.
Matching statistic: St000149
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(load all 2 compositions to match this statistic)
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000149: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000149: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [2]
=> [2]
=> 1
1 => [1,1] => [1,1]
=> [1,1]
=> 0
00 => [3] => [3]
=> [2,1]
=> 0
01 => [2,1] => [2,1]
=> [3]
=> 1
10 => [1,2] => [2,1]
=> [3]
=> 1
11 => [1,1,1] => [1,1,1]
=> [1,1,1]
=> 0
000 => [4] => [4]
=> [2,2]
=> 1
001 => [3,1] => [3,1]
=> [2,1,1]
=> 0
010 => [2,2] => [2,2]
=> [4]
=> 2
011 => [2,1,1] => [2,1,1]
=> [3,1]
=> 1
100 => [1,3] => [3,1]
=> [2,1,1]
=> 0
101 => [1,2,1] => [2,1,1]
=> [3,1]
=> 1
110 => [1,1,2] => [2,1,1]
=> [3,1]
=> 1
111 => [1,1,1,1] => [1,1,1,1]
=> [1,1,1,1]
=> 0
0000 => [5] => [5]
=> [2,2,1]
=> 0
0001 => [4,1] => [4,1]
=> [3,2]
=> 1
0010 => [3,2] => [3,2]
=> [4,1]
=> 1
0011 => [3,1,1] => [3,1,1]
=> [2,1,1,1]
=> 0
0100 => [2,3] => [3,2]
=> [4,1]
=> 1
0101 => [2,2,1] => [2,2,1]
=> [5]
=> 2
0110 => [2,1,2] => [2,2,1]
=> [5]
=> 2
0111 => [2,1,1,1] => [2,1,1,1]
=> [3,1,1]
=> 1
1000 => [1,4] => [4,1]
=> [3,2]
=> 1
1001 => [1,3,1] => [3,1,1]
=> [2,1,1,1]
=> 0
1010 => [1,2,2] => [2,2,1]
=> [5]
=> 2
1011 => [1,2,1,1] => [2,1,1,1]
=> [3,1,1]
=> 1
1100 => [1,1,3] => [3,1,1]
=> [2,1,1,1]
=> 0
1101 => [1,1,2,1] => [2,1,1,1]
=> [3,1,1]
=> 1
1110 => [1,1,1,2] => [2,1,1,1]
=> [3,1,1]
=> 1
1111 => [1,1,1,1,1] => [1,1,1,1,1]
=> [1,1,1,1,1]
=> 0
00000 => [6] => [6]
=> [2,2,2]
=> 1
00001 => [5,1] => [5,1]
=> [2,2,1,1]
=> 0
00010 => [4,2] => [4,2]
=> [4,2]
=> 2
00011 => [4,1,1] => [4,1,1]
=> [4,1,1]
=> 1
00100 => [3,3] => [3,3]
=> [3,2,1]
=> 0
00101 => [3,2,1] => [3,2,1]
=> [3,3]
=> 1
00110 => [3,1,2] => [3,2,1]
=> [3,3]
=> 1
00111 => [3,1,1,1] => [3,1,1,1]
=> [2,1,1,1,1]
=> 0
01000 => [2,4] => [4,2]
=> [4,2]
=> 2
01001 => [2,3,1] => [3,2,1]
=> [3,3]
=> 1
01010 => [2,2,2] => [2,2,2]
=> [6]
=> 3
01011 => [2,2,1,1] => [2,2,1,1]
=> [5,1]
=> 2
01100 => [2,1,3] => [3,2,1]
=> [3,3]
=> 1
01101 => [2,1,2,1] => [2,2,1,1]
=> [5,1]
=> 2
01110 => [2,1,1,2] => [2,2,1,1]
=> [5,1]
=> 2
01111 => [2,1,1,1,1] => [2,1,1,1,1]
=> [3,1,1,1]
=> 1
10000 => [1,5] => [5,1]
=> [2,2,1,1]
=> 0
10001 => [1,4,1] => [4,1,1]
=> [4,1,1]
=> 1
10010 => [1,3,2] => [3,2,1]
=> [3,3]
=> 1
10011 => [1,3,1,1] => [3,1,1,1]
=> [2,1,1,1,1]
=> 0
Description
The number of cells of the partition whose leg is zero and arm is odd.
This statistic is equidistributed with [[St000143]], see [1].
Matching statistic: St000150
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(load all 2 compositions to match this statistic)
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000150: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000150: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [2]
=> [1,1]
=> 1
1 => [1,1] => [1,1]
=> [2]
=> 0
00 => [3] => [3]
=> [3]
=> 0
01 => [2,1] => [2,1]
=> [1,1,1]
=> 1
10 => [1,2] => [2,1]
=> [1,1,1]
=> 1
11 => [1,1,1] => [1,1,1]
=> [2,1]
=> 0
000 => [4] => [4]
=> [2,2]
=> 1
001 => [3,1] => [3,1]
=> [3,1]
=> 0
010 => [2,2] => [2,2]
=> [1,1,1,1]
=> 2
011 => [2,1,1] => [2,1,1]
=> [2,1,1]
=> 1
100 => [1,3] => [3,1]
=> [3,1]
=> 0
101 => [1,2,1] => [2,1,1]
=> [2,1,1]
=> 1
110 => [1,1,2] => [2,1,1]
=> [2,1,1]
=> 1
111 => [1,1,1,1] => [1,1,1,1]
=> [4]
=> 0
0000 => [5] => [5]
=> [5]
=> 0
0001 => [4,1] => [4,1]
=> [2,2,1]
=> 1
0010 => [3,2] => [3,2]
=> [3,1,1]
=> 1
0011 => [3,1,1] => [3,1,1]
=> [3,2]
=> 0
0100 => [2,3] => [3,2]
=> [3,1,1]
=> 1
0101 => [2,2,1] => [2,2,1]
=> [1,1,1,1,1]
=> 2
0110 => [2,1,2] => [2,2,1]
=> [1,1,1,1,1]
=> 2
0111 => [2,1,1,1] => [2,1,1,1]
=> [2,1,1,1]
=> 1
1000 => [1,4] => [4,1]
=> [2,2,1]
=> 1
1001 => [1,3,1] => [3,1,1]
=> [3,2]
=> 0
1010 => [1,2,2] => [2,2,1]
=> [1,1,1,1,1]
=> 2
1011 => [1,2,1,1] => [2,1,1,1]
=> [2,1,1,1]
=> 1
1100 => [1,1,3] => [3,1,1]
=> [3,2]
=> 0
1101 => [1,1,2,1] => [2,1,1,1]
=> [2,1,1,1]
=> 1
1110 => [1,1,1,2] => [2,1,1,1]
=> [2,1,1,1]
=> 1
1111 => [1,1,1,1,1] => [1,1,1,1,1]
=> [4,1]
=> 0
00000 => [6] => [6]
=> [3,3]
=> 1
00001 => [5,1] => [5,1]
=> [5,1]
=> 0
00010 => [4,2] => [4,2]
=> [2,2,1,1]
=> 2
00011 => [4,1,1] => [4,1,1]
=> [2,2,2]
=> 1
00100 => [3,3] => [3,3]
=> [6]
=> 0
00101 => [3,2,1] => [3,2,1]
=> [3,1,1,1]
=> 1
00110 => [3,1,2] => [3,2,1]
=> [3,1,1,1]
=> 1
00111 => [3,1,1,1] => [3,1,1,1]
=> [3,2,1]
=> 0
01000 => [2,4] => [4,2]
=> [2,2,1,1]
=> 2
01001 => [2,3,1] => [3,2,1]
=> [3,1,1,1]
=> 1
01010 => [2,2,2] => [2,2,2]
=> [1,1,1,1,1,1]
=> 3
01011 => [2,2,1,1] => [2,2,1,1]
=> [2,1,1,1,1]
=> 2
01100 => [2,1,3] => [3,2,1]
=> [3,1,1,1]
=> 1
01101 => [2,1,2,1] => [2,2,1,1]
=> [2,1,1,1,1]
=> 2
01110 => [2,1,1,2] => [2,2,1,1]
=> [2,1,1,1,1]
=> 2
01111 => [2,1,1,1,1] => [2,1,1,1,1]
=> [4,1,1]
=> 1
10000 => [1,5] => [5,1]
=> [5,1]
=> 0
10001 => [1,4,1] => [4,1,1]
=> [2,2,2]
=> 1
10010 => [1,3,2] => [3,2,1]
=> [3,1,1,1]
=> 1
10011 => [1,3,1,1] => [3,1,1,1]
=> [3,2,1]
=> 0
Description
The floored half-sum of the multiplicities of a partition.
This statistic is equidistributed with [[St000143]] and [[St000149]], see [1].
Matching statistic: St000237
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000237: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> [2,1] => 1
1 => [1,1] => [1,0,1,0]
=> [1,2] => 0
00 => [3] => [1,1,1,0,0,0]
=> [3,2,1] => 0
01 => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 1
10 => [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0
000 => [4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => 0
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,6,5] => 2
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,3,2,1,5,6] => 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,2,1,6,5,4] => 0
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,2,1,5,4,6] => 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,2,1,4,6,5] => 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6] => 0
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,5,4,3] => 2
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,4,3,6] => 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5] => 3
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,5,4] => 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 2
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 0
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,3,2,6,5] => 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => 0
Description
The number of small exceedances.
This is the number of indices i such that πi=i+1.
Matching statistic: St001230
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001230: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001230: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 1
1 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 0
00 => [3] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 0
01 => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
10 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 0
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 0
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> 2
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 0
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,1,0,0]
=> 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 0
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 2
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 3
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,1,0,0,0]
=> 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,1,0,0]
=> 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 2
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 0
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 0
Description
The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property.
Matching statistic: St001465
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001465: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001465: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> [2,1] => 1
1 => [1,1] => [1,0,1,0]
=> [1,2] => 0
00 => [3] => [1,1,1,0,0,0]
=> [3,2,1] => 0
01 => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 1
10 => [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0
000 => [4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 0
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 0
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 0
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => 0
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,6,5] => 2
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,3,2,1,5,6] => 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,2,1,6,5,4] => 0
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,2,1,5,4,6] => 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,2,1,4,6,5] => 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6] => 0
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,5,4,3] => 2
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,4,3,6] => 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5] => 3
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,5,4] => 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => 2
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 2
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 0
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,3,2,6,5] => 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => 0
Description
The number of adjacent transpositions in the cycle decomposition of a permutation.
Matching statistic: St000137
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000137: Integer partitions ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 75%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000137: Integer partitions ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 75%
Values
0 => [1] => [[1],[]]
=> []
=> ? ∊ {0,1}
1 => [1] => [[1],[]]
=> []
=> ? ∊ {0,1}
00 => [2] => [[2],[]]
=> []
=> ? ∊ {0,0,1,1}
01 => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,1,1}
10 => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,1,1}
11 => [2] => [[2],[]]
=> []
=> ? ∊ {0,0,1,1}
000 => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,1,1,2}
001 => [2,1] => [[2,2],[1]]
=> [1]
=> 1
010 => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,2}
011 => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,2}
100 => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,2}
101 => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,2}
110 => [2,1] => [[2,2],[1]]
=> [1]
=> 1
111 => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,1,1,2}
0000 => [4] => [[4],[]]
=> []
=> ? ∊ {0,1,1,1,1,2,2,2}
0001 => [3,1] => [[3,3],[2]]
=> [2]
=> 0
0010 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
0011 => [2,2] => [[3,2],[1]]
=> [1]
=> 1
0100 => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,2,2,2}
0101 => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,2,2,2}
0110 => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
0111 => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,2,2,2}
1000 => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,2,2,2}
1001 => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
1010 => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,2,2,2}
1011 => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,2,2,2}
1100 => [2,2] => [[3,2],[1]]
=> [1]
=> 1
1101 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
1110 => [3,1] => [[3,3],[2]]
=> [2]
=> 0
1111 => [4] => [[4],[]]
=> []
=> ? ∊ {0,1,1,1,1,2,2,2}
00000 => [5] => [[5],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,3}
00001 => [4,1] => [[4,4],[3]]
=> [3]
=> 1
00010 => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
00011 => [3,2] => [[4,3],[2]]
=> [2]
=> 0
00100 => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
00101 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
00110 => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
00111 => [2,3] => [[4,2],[1]]
=> [1]
=> 1
01000 => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,3}
01001 => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 1
01010 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,3}
01011 => [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,3}
01100 => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 1
01101 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
01110 => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
01111 => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,3}
10000 => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,3}
10001 => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
10010 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
10011 => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 1
10100 => [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,3}
10101 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,3}
10110 => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 1
10111 => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,3}
11000 => [2,3] => [[4,2],[1]]
=> [1]
=> 1
11001 => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
11010 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
11011 => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
11100 => [3,2] => [[4,3],[2]]
=> [2]
=> 0
11101 => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
11110 => [4,1] => [[4,4],[3]]
=> [3]
=> 1
11111 => [5] => [[5],[]]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,3}
Description
The Grundy value of an integer partition.
Consider the two-player game on an integer partition.
In each move, a player removes either a box, or a 2x2-configuration of boxes such that the resulting diagram is still a partition.
The first player that cannot move lose. This happens exactly when the empty partition is reached.
The grundy value of an integer partition is defined as the grundy value of this two-player game as defined in [1].
This game was described to me during Norcom 2013, by Urban Larsson, and it seems to be quite difficult to give a good description of the partitions with Grundy value 0.
Matching statistic: St000621
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000621: Integer partitions ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 75%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000621: Integer partitions ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 75%
Values
0 => [2] => [[2],[]]
=> []
=> ? ∊ {0,1}
1 => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,1}
00 => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,1,1}
01 => [2,1] => [[2,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1}
10 => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,1,1}
11 => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1}
000 => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,1,1,1,2}
001 => [3,1] => [[3,3],[2]]
=> [2]
=> 1
010 => [2,2] => [[3,2],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,2}
011 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 0
100 => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,2}
101 => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> ? ∊ {0,0,1,1,1,2}
110 => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,2}
111 => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,1,1,1,2}
0000 => [5] => [[5],[]]
=> []
=> ? ∊ {0,1,1,1,1,2,2,2}
0001 => [4,1] => [[4,4],[3]]
=> [3]
=> 0
0010 => [3,2] => [[4,3],[2]]
=> [2]
=> 1
0011 => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1
0100 => [2,3] => [[4,2],[1]]
=> [1]
=> ? ∊ {0,1,1,1,1,2,2,2}
0101 => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
0110 => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 0
0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
1000 => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,2,2,2}
1001 => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 1
1010 => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> ? ∊ {0,1,1,1,1,2,2,2}
1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 0
1100 => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,2,2,2}
1101 => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> ? ∊ {0,1,1,1,1,2,2,2}
1110 => [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,2,2,2}
1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,1,1,1,1,2,2,2}
00000 => [6] => [[6],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,3}
00001 => [5,1] => [[5,5],[4]]
=> [4]
=> 1
00010 => [4,2] => [[5,4],[3]]
=> [3]
=> 0
00011 => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 2
00100 => [3,3] => [[5,3],[2]]
=> [2]
=> 1
00101 => [3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 2
00110 => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 1
00111 => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 2
01000 => [2,4] => [[5,2],[1]]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,3}
01001 => [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 1
01010 => [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 1
01011 => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> 2
01100 => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 0
01101 => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 1
01110 => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 0
01111 => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 0
10000 => [1,5] => [[5,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,3}
10001 => [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 0
10010 => [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 1
10011 => [1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 1
10100 => [1,2,3] => [[4,2,1],[1]]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,3}
10101 => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 1
10110 => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 0
10111 => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 0
11000 => [1,1,4] => [[4,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,3}
11001 => [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 1
11010 => [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,3}
11011 => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 0
11100 => [1,1,1,3] => [[3,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,3}
11101 => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,3}
11110 => [1,1,1,1,2] => [[2,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,3}
11111 => [1,1,1,1,1,1] => [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,3}
Description
The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even.
To be precise, this is given for a partition λ⊢n by the number of standard tableaux T of shape λ such that min is even.
This notion was used in [1, Proposition 2.3], see also [2, Theorem 1.1].
The case of an odd minimum is [[St000620]].
Matching statistic: St000755
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000755: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 52%●distinct values known / distinct values provided: 50%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000755: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 52%●distinct values known / distinct values provided: 50%
Values
0 => [1] => [[1],[]]
=> []
=> ? ∊ {0,1}
1 => [1] => [[1],[]]
=> []
=> ? ∊ {0,1}
00 => [2] => [[2],[]]
=> []
=> ? ∊ {0,0,1,1}
01 => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,1,1}
10 => [1,1] => [[1,1],[]]
=> []
=> ? ∊ {0,0,1,1}
11 => [2] => [[2],[]]
=> []
=> ? ∊ {0,0,1,1}
000 => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,1,1,2}
001 => [2,1] => [[2,2],[1]]
=> [1]
=> 1
010 => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,2}
011 => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,2}
100 => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,2}
101 => [1,1,1] => [[1,1,1],[]]
=> []
=> ? ∊ {0,0,0,1,1,2}
110 => [2,1] => [[2,2],[1]]
=> [1]
=> 1
111 => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,0,1,1,2}
0000 => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2}
0001 => [3,1] => [[3,3],[2]]
=> [2]
=> 2
0010 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
0011 => [2,2] => [[3,2],[1]]
=> [1]
=> 1
0100 => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2}
0101 => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2}
0110 => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
0111 => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2}
1000 => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2}
1001 => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1
1010 => [1,1,1,1] => [[1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2}
1011 => [1,1,2] => [[2,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2}
1100 => [2,2] => [[3,2],[1]]
=> [1]
=> 1
1101 => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1
1110 => [3,1] => [[3,3],[2]]
=> [2]
=> 2
1111 => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,2}
00000 => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,3}
00001 => [4,1] => [[4,4],[3]]
=> [3]
=> 1
00010 => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
00011 => [3,2] => [[4,3],[2]]
=> [2]
=> 2
00100 => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
00101 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
00110 => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 2
00111 => [2,3] => [[4,2],[1]]
=> [1]
=> 1
01000 => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,3}
01001 => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 1
01010 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,3}
01011 => [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,3}
01100 => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 1
01101 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
01110 => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 2
01111 => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,3}
10000 => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,3}
10001 => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 2
10010 => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
10011 => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 1
10100 => [1,1,1,2] => [[2,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,3}
10101 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,3}
10110 => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 1
10111 => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,3}
11000 => [2,3] => [[4,2],[1]]
=> [1]
=> 1
11001 => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 2
11010 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
11011 => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1
11100 => [3,2] => [[4,3],[2]]
=> [2]
=> 2
11101 => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2
11110 => [4,1] => [[4,4],[3]]
=> [3]
=> 1
11111 => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,1,3}
Description
The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition.
Consider the recurrence f(n)=\sum_{p\in\lambda} f(n-p). This statistic returns the number of distinct real roots of the associated characteristic polynomial.
For example, the partition (2,1) corresponds to the recurrence f(n)=f(n-1)+f(n-2) with associated characteristic polynomial x^2-x-1, which has two real roots.
The following 24 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001568The smallest positive integer that does not appear twice in the partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000934The 2-degree of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000455The second largest eigenvalue of a graph if it is integral. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition.
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