Your data matches 193 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000782
(load all 2 compositions to match this statistic)
Mapping
St000782: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2),(3,4),(5,6)]
 => 1
[(1,3),(2,4),(5,6)]
 => 1
[(1,4),(2,3),(5,6)]
 => 1
[(1,5),(2,3),(4,6)]
 => 1
[(1,6),(2,3),(4,5)]
 => 1
[(1,6),(2,4),(3,5)]
 => 0
[(1,5),(2,4),(3,6)]
 => 1
[(1,4),(2,5),(3,6)]
 => 0
[(1,3),(2,5),(4,6)]
 => 0
[(1,2),(3,5),(4,6)]
 => 1
[(1,2),(3,6),(4,5)]
 => 1
[(1,3),(2,6),(4,5)]
 => 1
[(1,4),(2,6),(3,5)]
 => 1
[(1,5),(2,6),(3,4)]
 => 1
[(1,6),(2,5),(3,4)]
 => 1
Description
The indicator function of whether a given perfect matching is an L & P matching. An L&P matching is built inductively as follows: starting with either a single edge, or a hairpin $([1,3],[2,4])$, insert a noncrossing matching or inflate an edge by a ladder, that is, a number of nested edges. The number of L&P matchings is (see [thm. 1, 2]) $$\frac{1}{2} \cdot 4^{n} + \frac{1}{n + 1}{2 \, n \choose n} - {2 \, n + 1 \choose n} + {2 \, n - 1 \choose n - 1}$$
Matching statistic: St001159
Mapping
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001159: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [6,4,2,1,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
[(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => [6,1,3,4,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
[(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => [6,4,3,2,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
[(1,5),(2,3),(4,6)]
 => [5,3,2,6,1,4] => [3,5,2,1,6,4] => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 1
[(1,6),(2,3),(4,5)]
 => [6,3,2,5,4,1] => [6,3,5,2,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
[(1,6),(2,4),(3,5)]
 => [6,4,5,2,3,1] => [2,4,6,5,3,1] => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 0
[(1,5),(2,4),(3,6)]
 => [5,4,6,2,1,3] => [2,5,4,6,1,3] => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 1
[(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => [1,2,4,5,6,3] => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 0
[(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => [3,1,5,6,2,4] => [1,1,1,0,0,1,1,0,1,0,0,0]
 => 0
[(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => [2,5,1,3,6,4] => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 1
[(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => [6,5,4,2,1,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
[(1,3),(2,6),(4,5)]
 => [3,6,1,5,4,2] => [6,3,5,1,4,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
[(1,4),(2,6),(3,5)]
 => [4,6,5,1,3,2] => [6,1,4,5,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
[(1,5),(2,6),(3,4)]
 => [5,6,4,3,1,2] => [1,5,6,4,3,2] => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 1
[(1,6),(2,5),(3,4)]
 => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => 1
Description
Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra.
Matching statistic: St000994
Mapping
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000994: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [6,4,2,1,3,5] => [6,5,3,2,4,1] => 2 = 1 + 1
[(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => [6,1,3,4,2,5] => [3,4,6,5,2,1] => 2 = 1 + 1
[(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => [6,4,3,2,1,5] => [3,4,2,6,5,1] => 2 = 1 + 1
[(1,5),(2,3),(4,6)]
 => [5,3,2,6,1,4] => [3,5,2,1,6,4] => [6,4,1,3,2,5] => 2 = 1 + 1
[(1,6),(2,3),(4,5)]
 => [6,3,2,5,4,1] => [6,3,5,2,4,1] => [5,4,2,3,6,1] => 2 = 1 + 1
[(1,6),(2,4),(3,5)]
 => [6,4,5,2,3,1] => [2,4,6,5,3,1] => [6,1,2,4,5,3] => 1 = 0 + 1
[(1,5),(2,4),(3,6)]
 => [5,4,6,2,1,3] => [2,5,4,6,1,3] => [5,1,2,6,3,4] => 2 = 1 + 1
[(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => [1,2,4,5,6,3] => [1,2,6,3,4,5] => 1 = 0 + 1
[(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => [3,1,5,6,2,4] => [5,2,1,3,6,4] => 1 = 0 + 1
[(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => [2,5,1,3,6,4] => [6,4,3,1,2,5] => 2 = 1 + 1
[(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => [6,5,4,2,1,3] => [6,3,4,2,5,1] => 2 = 1 + 1
[(1,3),(2,6),(4,5)]
 => [3,6,1,5,4,2] => [6,3,5,1,4,2] => [6,2,3,5,4,1] => 2 = 1 + 1
[(1,4),(2,6),(3,5)]
 => [4,6,5,1,3,2] => [6,1,4,5,3,2] => [5,3,4,6,2,1] => 2 = 1 + 1
[(1,5),(2,6),(3,4)]
 => [5,6,4,3,1,2] => [1,5,6,4,3,2] => [1,4,6,2,5,3] => 2 = 1 + 1
[(1,6),(2,5),(3,4)]
 => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [4,3,5,2,6,1] => 2 = 1 + 1
Description
The number of cycle peaks and the number of cycle valleys of a permutation. A '''cycle peak''' of a permutation $\pi$ is an index $i$ such that $\pi^{-1}(i) < i > \pi(i)$. Analogously, a '''cycle valley''' is an index $i$ such that $\pi^{-1}(i) > i < \pi(i)$. Clearly, every cycle of $\pi$ contains as many peaks as valleys.
Matching statistic: St001928
Mapping
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
St001928: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [2,1,4,3,5] => [3,2,5,4,1] => 2 = 1 + 1
[(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => [3,4,1,2,5] => [4,5,2,3,1] => 2 = 1 + 1
[(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => [4,3,2,1,5] => [5,4,3,2,1] => 2 = 1 + 1
[(1,5),(2,3),(4,6)]
 => [5,3,2,6,1,4] => [5,3,2,1,4] => [5,4,3,1,2] => 2 = 1 + 1
[(1,6),(2,3),(4,5)]
 => [6,3,2,5,4,1] => [3,2,5,4,1] => [1,3,2,5,4] => 2 = 1 + 1
[(1,6),(2,4),(3,5)]
 => [6,4,5,2,3,1] => [4,5,2,3,1] => [1,4,5,2,3] => 1 = 0 + 1
[(1,5),(2,4),(3,6)]
 => [5,4,6,2,1,3] => [5,4,2,1,3] => [5,4,1,3,2] => 2 = 1 + 1
[(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => [4,5,1,2,3] => [4,5,1,2,3] => 1 = 0 + 1
[(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => [3,5,1,2,4] => [4,5,2,1,3] => 1 = 0 + 1
[(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => [2,1,5,3,4] => [3,2,5,1,4] => 2 = 1 + 1
[(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => [2,1,5,4,3] => [3,2,1,5,4] => 2 = 1 + 1
[(1,3),(2,6),(4,5)]
 => [3,6,1,5,4,2] => [3,1,5,4,2] => [3,1,2,5,4] => 2 = 1 + 1
[(1,4),(2,6),(3,5)]
 => [4,6,5,1,3,2] => [4,5,1,3,2] => [4,1,5,2,3] => 2 = 1 + 1
[(1,5),(2,6),(3,4)]
 => [5,6,4,3,1,2] => [5,4,3,1,2] => [5,1,4,3,2] => 2 = 1 + 1
[(1,6),(2,5),(3,4)]
 => [6,5,4,3,2,1] => [5,4,3,2,1] => [1,5,4,3,2] => 2 = 1 + 1
Description
The number of non-overlapping descents in a permutation. In other words, any maximal descending subsequence $\pi_i,\pi_{i+1},\dots,\pi_k$ contributes $\lfloor\frac{k-i+1}{2}\rfloor$ to the total count.
Matching statistic: St000069
(load all 2 compositions to match this statistic)
Mapping
Mp00283: Perfect matchings —non-nesting-exceedence permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000069: Posets ⟶ ℤResult quality: 50% ●values known / values provided: 67%●distinct values known / distinct values provided: 50%
Values
[(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [2,1,4,3,6,5] => ([(0,1),(0,2),(0,3),(1,5),(1,6),(1,12),(2,6),(2,7),(2,12),(3,5),(3,7),(3,12),(5,9),(5,10),(6,9),(6,11),(7,9),(7,10),(7,11),(8,4),(9,13),(10,8),(10,13),(11,8),(11,13),(12,10),(12,11),(13,4)],14)
 => 1
[(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => [3,1,4,2,6,5] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,10),(1,15),(1,16),(2,11),(2,15),(2,16),(3,13),(3,15),(3,16),(4,12),(4,15),(4,16),(5,8),(5,9),(5,14),(5,20),(6,5),(6,10),(6,11),(6,12),(6,13),(8,17),(8,19),(9,17),(9,19),(10,18),(10,20),(11,14),(11,18),(11,20),(12,8),(12,18),(12,20),(13,9),(13,18),(13,20),(14,17),(14,19),(15,14),(15,20),(16,14),(16,18),(17,7),(18,19),(19,7),(20,17),(20,19)],21)
 => 1
[(1,4),(2,3),(5,6)]
 => [3,4,2,1,6,5] => [4,1,3,2,6,5] => ([(0,1),(0,3),(0,4),(0,5),(1,11),(1,14),(2,6),(2,8),(2,15),(3,10),(3,12),(3,14),(4,9),(4,10),(4,14),(5,2),(5,9),(5,11),(5,12),(6,17),(6,18),(8,17),(9,13),(9,15),(9,16),(10,13),(10,15),(11,8),(11,16),(12,6),(12,13),(12,16),(13,18),(14,15),(14,16),(15,17),(15,18),(16,17),(16,18),(17,7),(18,7)],19)
 => 1
[(1,5),(2,3),(4,6)]
 => [3,5,2,6,1,4] => [5,1,3,2,6,4] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,21),(1,22),(2,8),(2,10),(2,17),(3,12),(3,13),(3,15),(3,17),(4,11),(4,14),(4,15),(4,17),(5,8),(5,9),(5,13),(5,14),(6,9),(6,10),(6,11),(6,12),(8,19),(8,23),(9,1),(9,19),(9,20),(9,23),(10,18),(10,19),(11,18),(11,20),(12,18),(12,20),(12,23),(13,16),(13,23),(14,16),(14,20),(14,23),(15,16),(15,19),(15,20),(16,22),(17,18),(17,19),(17,23),(18,21),(19,21),(19,22),(20,21),(20,22),(21,7),(22,7),(23,21),(23,22)],24)
 => 1
[(1,6),(2,3),(4,5)]
 => [3,5,2,6,4,1] => [6,1,3,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(1,10),(1,15),(2,8),(2,11),(2,15),(3,7),(3,8),(3,9),(4,7),(4,10),(4,11),(4,15),(5,17),(7,12),(7,13),(7,16),(8,13),(8,16),(9,12),(9,16),(10,12),(10,14),(11,5),(11,13),(11,14),(12,18),(13,17),(13,18),(14,17),(14,18),(15,5),(15,14),(15,16),(16,17),(16,18),(17,6),(18,6)],19)
 => 1
[(1,6),(2,4),(3,5)]
 => [4,5,6,2,3,1] => [6,1,4,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,19),(1,21),(2,13),(2,15),(2,19),(2,21),(3,12),(3,14),(3,19),(3,21),(4,10),(4,11),(4,19),(4,21),(5,9),(5,11),(5,14),(5,15),(5,21),(6,8),(6,9),(6,10),(6,12),(6,13),(8,20),(8,24),(9,16),(9,17),(9,24),(9,25),(10,20),(10,24),(10,25),(11,18),(11,25),(12,16),(12,20),(12,24),(13,17),(13,20),(13,24),(14,16),(14,18),(14,25),(15,17),(15,18),(15,25),(16,22),(16,23),(17,22),(17,23),(18,23),(19,20),(19,25),(20,22),(21,18),(21,24),(21,25),(22,7),(23,7),(24,22),(24,23),(25,22),(25,23)],26)
 => ? = 0
[(1,5),(2,4),(3,6)]
 => [4,5,6,2,1,3] => [5,1,4,2,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(1,12),(1,18),(1,20),(2,11),(2,13),(2,17),(2,18),(3,14),(3,15),(3,17),(3,18),(3,20),(4,9),(4,13),(4,15),(4,20),(5,7),(5,9),(5,12),(5,17),(6,7),(6,10),(6,11),(6,14),(6,20),(7,24),(7,25),(7,26),(9,19),(9,26),(10,24),(10,25),(11,21),(11,24),(11,25),(12,24),(12,26),(13,19),(13,21),(14,16),(14,24),(14,25),(15,16),(15,19),(15,21),(16,22),(16,23),(17,19),(17,25),(17,26),(18,21),(18,24),(18,26),(19,23),(20,16),(20,21),(20,25),(20,26),(21,22),(21,23),(22,8),(23,8),(24,22),(25,22),(25,23),(26,22),(26,23)],27)
 => ? = 1
[(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => [4,1,5,2,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,12),(1,16),(1,17),(1,24),(2,11),(2,15),(2,17),(2,24),(3,9),(3,13),(3,15),(3,24),(4,10),(4,14),(4,16),(4,24),(5,7),(5,9),(5,11),(5,14),(5,24),(6,7),(6,10),(6,12),(6,13),(6,24),(7,21),(7,22),(7,25),(9,21),(9,25),(10,22),(10,25),(11,19),(11,21),(11,25),(12,20),(12,22),(12,25),(13,19),(13,22),(13,25),(14,20),(14,21),(14,25),(15,19),(15,25),(16,20),(16,25),(17,19),(17,20),(18,8),(19,18),(19,23),(20,18),(20,23),(21,18),(21,23),(22,18),(22,23),(23,8),(24,19),(24,20),(24,21),(24,22),(25,23)],26)
 => ? = 0
[(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => [3,1,5,2,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,15),(1,19),(2,12),(2,14),(2,19),(3,9),(3,11),(3,14),(3,16),(3,19),(4,10),(4,11),(4,15),(4,16),(4,19),(5,7),(5,9),(5,12),(5,15),(5,16),(6,7),(6,10),(6,13),(6,14),(6,16),(7,22),(7,23),(7,24),(9,17),(9,22),(9,24),(10,18),(10,23),(10,24),(11,17),(11,18),(11,22),(11,23),(12,22),(12,24),(13,23),(13,24),(14,17),(14,23),(14,24),(15,18),(15,22),(15,24),(16,17),(16,18),(16,22),(16,23),(17,20),(17,21),(18,20),(18,21),(19,22),(19,23),(20,8),(21,8),(22,20),(22,21),(23,20),(23,21),(24,21)],25)
 => ? = 0
[(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => [2,1,5,3,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,6),(1,10),(1,15),(1,16),(2,11),(2,15),(2,16),(3,13),(3,15),(3,16),(4,12),(4,15),(4,16),(5,8),(5,9),(5,14),(5,20),(6,5),(6,10),(6,11),(6,12),(6,13),(8,17),(8,19),(9,17),(9,19),(10,18),(10,20),(11,14),(11,18),(11,20),(12,8),(12,18),(12,20),(13,9),(13,18),(13,20),(14,17),(14,19),(15,14),(15,20),(16,14),(16,18),(17,7),(18,19),(19,7),(20,17),(20,19)],21)
 => 1
[(1,2),(3,6),(4,5)]
 => [2,1,5,6,4,3] => [2,1,6,3,5,4] => ([(0,1),(0,3),(0,4),(0,5),(1,11),(1,15),(2,6),(2,8),(2,18),(3,12),(3,13),(3,15),(4,10),(4,13),(4,15),(5,2),(5,10),(5,11),(5,12),(6,16),(6,17),(7,16),(7,17),(8,16),(10,14),(10,18),(11,8),(11,18),(12,6),(12,14),(12,18),(13,7),(13,14),(14,17),(15,7),(15,18),(16,9),(17,9),(18,16),(18,17)],19)
 => 1
[(1,3),(2,6),(4,5)]
 => [3,5,1,6,4,2] => [3,1,6,2,5,4] => ([(0,1),(0,3),(0,4),(0,5),(0,6),(1,16),(1,18),(2,8),(2,17),(2,19),(2,22),(3,11),(3,12),(3,18),(4,13),(4,14),(4,16),(4,18),(5,10),(5,11),(5,13),(5,16),(6,2),(6,10),(6,12),(6,14),(6,18),(7,20),(8,20),(8,21),(10,15),(10,19),(10,22),(11,15),(11,22),(12,15),(12,17),(12,19),(13,7),(13,19),(13,22),(14,8),(14,17),(14,19),(14,22),(15,21),(16,7),(16,22),(17,20),(17,21),(18,17),(18,22),(19,20),(19,21),(20,9),(21,9),(22,20),(22,21)],23)
 => 1
[(1,4),(2,6),(3,5)]
 => [4,5,6,1,3,2] => [4,1,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,9),(1,18),(1,22),(2,11),(2,14),(2,16),(2,18),(3,9),(3,14),(3,15),(3,22),(4,12),(4,13),(4,16),(4,22),(5,10),(5,13),(5,15),(5,18),(5,22),(6,8),(6,10),(6,11),(6,12),(6,22),(8,20),(8,25),(9,19),(9,25),(10,20),(10,21),(10,25),(10,26),(11,17),(11,25),(11,26),(12,17),(12,20),(12,26),(13,21),(13,26),(14,19),(14,26),(15,19),(15,21),(15,25),(16,17),(16,26),(17,24),(18,19),(18,25),(18,26),(19,23),(20,23),(20,24),(21,23),(21,24),(22,20),(22,21),(22,25),(22,26),(23,7),(24,7),(25,23),(25,24),(26,23),(26,24)],27)
 => ? = 1
[(1,5),(2,6),(3,4)]
 => [4,5,6,3,1,2] => [6,2,5,1,4,3] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,18),(1,21),(2,11),(2,12),(2,19),(3,9),(3,13),(3,19),(4,10),(4,13),(4,19),(5,10),(5,12),(5,14),(5,19),(6,1),(6,9),(6,11),(6,14),(6,19),(8,17),(8,20),(9,18),(9,21),(10,16),(10,21),(11,15),(11,18),(11,21),(12,15),(12,16),(13,21),(14,8),(14,15),(14,16),(14,18),(15,17),(15,20),(16,17),(16,20),(17,7),(18,17),(18,20),(19,16),(19,18),(19,21),(20,7),(21,20)],22)
 => 1
[(1,6),(2,5),(3,4)]
 => [4,5,6,3,2,1] => [5,2,6,1,4,3] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,17),(1,18),(1,21),(2,11),(2,13),(2,15),(3,12),(3,14),(3,15),(4,8),(4,10),(4,12),(4,15),(5,9),(5,10),(5,13),(5,15),(6,1),(6,8),(6,9),(6,11),(6,14),(8,17),(8,18),(8,19),(8,20),(9,17),(9,18),(9,19),(9,20),(10,20),(10,21),(11,17),(11,19),(12,18),(12,20),(13,17),(13,21),(14,18),(14,19),(14,21),(15,19),(15,20),(15,21),(16,7),(17,16),(17,22),(18,16),(18,22),(19,16),(19,22),(20,16),(20,22),(21,22),(22,7)],23)
 => 1
Description
The number of maximal elements of a poset.
Matching statistic: St001260
(load all 36 compositions to match this statistic)
Mapping
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St001260: Alternating sign matrices ⟶ ℤResult quality: 50% ●values known / values provided: 53%●distinct values known / distinct values provided: 50%
Values
[(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [3,2,5,4,1,6] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
 => 1
[(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => [4,5,2,3,1,6] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
 => 1
[(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => [5,4,3,2,1,6] => [[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1]]
 => 1
[(1,5),(2,3),(4,6)]
 => [5,3,2,6,1,4] => [6,4,3,1,5,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0]]
 => ? = 1
[(1,6),(2,3),(4,5)]
 => [6,3,2,5,4,1] => [1,5,4,2,3,6] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
 => 1
[(1,6),(2,4),(3,5)]
 => [6,4,5,2,3,1] => [1,6,2,5,3,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0]]
 => ? = 0
[(1,5),(2,4),(3,6)]
 => [5,4,6,2,1,3] => [6,5,1,4,3,2] => [[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 1
[(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => [5,6,1,3,4,2] => [[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0]]
 => ? = 0
[(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => [4,6,2,1,5,3] => [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
 => ? = 0
[(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => [3,2,6,1,5,4] => [[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0]]
 => ? = 1
[(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => [3,2,1,4,5,6] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => 1
[(1,3),(2,6),(4,5)]
 => [3,6,1,5,4,2] => [4,1,3,2,5,6] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => 1
[(1,4),(2,6),(3,5)]
 => [4,6,5,1,3,2] => [5,1,2,4,3,6] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1]]
 => 1
[(1,5),(2,6),(3,4)]
 => [5,6,4,3,1,2] => [6,1,2,3,5,4] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0]]
 => ? = 1
[(1,6),(2,5),(3,4)]
 => [6,5,4,3,2,1] => [1,2,3,4,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => 1
Description
The permanent of an alternating sign matrix.
Matching statistic: St001925
(load all 33 compositions to match this statistic)
Mapping
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St001925: Alternating sign matrices ⟶ ℤResult quality: 50% ●values known / values provided: 53%●distinct values known / distinct values provided: 50%
Values
[(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [3,2,5,4,1,6] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
 => 5 = 1 + 4
[(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => [4,5,2,3,1,6] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
 => 5 = 1 + 4
[(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => [5,4,3,2,1,6] => [[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1]]
 => 5 = 1 + 4
[(1,5),(2,3),(4,6)]
 => [5,3,2,6,1,4] => [6,4,3,1,5,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0]]
 => ? = 1 + 4
[(1,6),(2,3),(4,5)]
 => [6,3,2,5,4,1] => [1,5,4,2,3,6] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
 => 5 = 1 + 4
[(1,6),(2,4),(3,5)]
 => [6,4,5,2,3,1] => [1,6,2,5,3,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0]]
 => ? = 0 + 4
[(1,5),(2,4),(3,6)]
 => [5,4,6,2,1,3] => [6,5,1,4,3,2] => [[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 1 + 4
[(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => [5,6,1,3,4,2] => [[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0]]
 => ? = 0 + 4
[(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => [4,6,2,1,5,3] => [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
 => ? = 0 + 4
[(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => [3,2,6,1,5,4] => [[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0]]
 => ? = 1 + 4
[(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => [3,2,1,4,5,6] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => 5 = 1 + 4
[(1,3),(2,6),(4,5)]
 => [3,6,1,5,4,2] => [4,1,3,2,5,6] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => 5 = 1 + 4
[(1,4),(2,6),(3,5)]
 => [4,6,5,1,3,2] => [5,1,2,4,3,6] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1]]
 => 5 = 1 + 4
[(1,5),(2,6),(3,4)]
 => [5,6,4,3,1,2] => [6,1,2,3,5,4] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0]]
 => ? = 1 + 4
[(1,6),(2,5),(3,4)]
 => [6,5,4,3,2,1] => [1,2,3,4,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => 5 = 1 + 4
Description
The minimal number of zeros in a row of an alternating sign matrix.
Matching statistic: St000890
(load all 33 compositions to match this statistic)
Mapping
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
St000890: Alternating sign matrices ⟶ ℤResult quality: 50% ●values known / values provided: 53%●distinct values known / distinct values provided: 50%
Values
[(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [3,2,5,4,1,6] => [[0,0,0,0,1,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1]]
 => 6 = 1 + 5
[(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => [4,5,2,3,1,6] => [[0,0,0,0,1,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
 => 6 = 1 + 5
[(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => [5,4,3,2,1,6] => [[0,0,0,0,1,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1]]
 => 6 = 1 + 5
[(1,5),(2,3),(4,6)]
 => [5,3,2,6,1,4] => [6,4,3,1,5,2] => [[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0]]
 => ? = 1 + 5
[(1,6),(2,3),(4,5)]
 => [6,3,2,5,4,1] => [1,5,4,2,3,6] => [[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,1,0,0,0],[0,1,0,0,0,0],[0,0,0,0,0,1]]
 => 6 = 1 + 5
[(1,6),(2,4),(3,5)]
 => [6,4,5,2,3,1] => [1,6,2,5,3,4] => [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,1,0,0,0,0]]
 => ? = 0 + 5
[(1,5),(2,4),(3,6)]
 => [5,4,6,2,1,3] => [6,5,1,4,3,2] => [[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0]]
 => ? = 1 + 5
[(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => [5,6,1,3,4,2] => [[0,0,1,0,0,0],[0,0,0,0,0,1],[0,0,0,1,0,0],[0,0,0,0,1,0],[1,0,0,0,0,0],[0,1,0,0,0,0]]
 => ? = 0 + 5
[(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => [4,6,2,1,5,3] => [[0,0,0,1,0,0],[0,0,1,0,0,0],[0,0,0,0,0,1],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,1,0,0,0,0]]
 => ? = 0 + 5
[(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => [3,2,6,1,5,4] => [[0,0,0,1,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[0,0,1,0,0,0]]
 => ? = 1 + 5
[(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => [3,2,1,4,5,6] => [[0,0,1,0,0,0],[0,1,0,0,0,0],[1,0,0,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => 6 = 1 + 5
[(1,3),(2,6),(4,5)]
 => [3,6,1,5,4,2] => [4,1,3,2,5,6] => [[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,0,0,0],[1,0,0,0,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => 6 = 1 + 5
[(1,4),(2,6),(3,5)]
 => [4,6,5,1,3,2] => [5,1,2,4,3,6] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,0,0],[1,0,0,0,0,0],[0,0,0,0,0,1]]
 => 6 = 1 + 5
[(1,5),(2,6),(3,4)]
 => [5,6,4,3,1,2] => [6,1,2,3,5,4] => [[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0],[1,0,0,0,0,0]]
 => ? = 1 + 5
[(1,6),(2,5),(3,4)]
 => [6,5,4,3,2,1] => [1,2,3,4,5,6] => [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
 => 6 = 1 + 5
Description
The number of nonzero entries in an alternating sign matrix.
Matching statistic: St001060
(load all 3 compositions to match this statistic)
Mapping
Mp00144: Perfect matchings —rotation⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001060: Graphs ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 50%
Values
[(1,2),(3,4),(5,6)]
 => [(1,6),(2,3),(4,5)]
 => [6,3,2,5,4,1] => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[(1,3),(2,4),(5,6)]
 => [(1,6),(2,4),(3,5)]
 => [6,4,5,2,3,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[(1,4),(2,3),(5,6)]
 => [(1,6),(2,5),(3,4)]
 => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[(1,5),(2,3),(4,6)]
 => [(1,5),(2,6),(3,4)]
 => [5,6,4,3,1,2] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[(1,6),(2,3),(4,5)]
 => [(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => ([(0,5),(1,4),(2,3)],6)
 => ? = 1 + 1
[(1,6),(2,4),(3,5)]
 => [(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 0 + 1
[(1,5),(2,4),(3,6)]
 => [(1,4),(2,6),(3,5)]
 => [4,6,5,1,3,2] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[(1,4),(2,5),(3,6)]
 => [(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
 => ? = 0 + 1
[(1,3),(2,5),(4,6)]
 => [(1,5),(2,4),(3,6)]
 => [5,4,6,2,1,3] => ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 0 + 1
[(1,2),(3,5),(4,6)]
 => [(1,5),(2,3),(4,6)]
 => [5,3,2,6,1,4] => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[(1,2),(3,6),(4,5)]
 => [(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[(1,3),(2,6),(4,5)]
 => [(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => ([(0,1),(2,4),(2,5),(3,4),(3,5)],6)
 => ? = 1 + 1
[(1,4),(2,6),(3,5)]
 => [(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
 => 2 = 1 + 1
[(1,5),(2,6),(3,4)]
 => [(1,3),(2,6),(4,5)]
 => [3,6,1,5,4,2] => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 2 = 1 + 1
[(1,6),(2,5),(3,4)]
 => [(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => ([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
Description
The distinguishing index of a graph. This is the smallest number of colours such that there is a colouring of the edges which is not preserved by any automorphism. If the graph has a connected component which is a single edge, or at least two isolated vertices, this statistic is undefined.
Matching statistic: St001199
(load all 2 compositions to match this statistic)
Mapping
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 100%
Values
[(1,2),(3,4),(5,6)]
 => [2,1,4,3,6,5] => [6,4,2,1,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[(1,3),(2,4),(5,6)]
 => [3,4,1,2,6,5] => [6,1,3,4,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[(1,4),(2,3),(5,6)]
 => [4,3,2,1,6,5] => [6,4,3,2,1,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[(1,5),(2,3),(4,6)]
 => [5,3,2,6,1,4] => [3,5,2,1,6,4] => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[(1,6),(2,3),(4,5)]
 => [6,3,2,5,4,1] => [6,3,5,2,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[(1,6),(2,4),(3,5)]
 => [6,4,5,2,3,1] => [2,4,6,5,3,1] => [1,1,0,1,1,0,1,1,0,0,0,0]
 => 1 = 0 + 1
[(1,5),(2,4),(3,6)]
 => [5,4,6,2,1,3] => [2,5,4,6,1,3] => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 2 = 1 + 1
[(1,4),(2,5),(3,6)]
 => [4,5,6,1,2,3] => [1,2,4,5,6,3] => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 1 = 0 + 1
[(1,3),(2,5),(4,6)]
 => [3,5,1,6,2,4] => [3,1,5,6,2,4] => [1,1,1,0,0,1,1,0,1,0,0,0]
 => 1 = 0 + 1
[(1,2),(3,5),(4,6)]
 => [2,1,5,6,3,4] => [2,5,1,3,6,4] => [1,1,0,1,1,1,0,0,0,1,0,0]
 => 2 = 1 + 1
[(1,2),(3,6),(4,5)]
 => [2,1,6,5,4,3] => [6,5,4,2,1,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[(1,3),(2,6),(4,5)]
 => [3,6,1,5,4,2] => [6,3,5,1,4,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[(1,4),(2,6),(3,5)]
 => [4,6,5,1,3,2] => [6,1,4,5,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
[(1,5),(2,6),(3,4)]
 => [5,6,4,3,1,2] => [1,5,6,4,3,2] => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 2 = 1 + 1
[(1,6),(2,5),(3,4)]
 => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1 + 1
Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
The following 183 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001271The competition number of a graph. St000065The number of entries equal to -1 in an alternating sign matrix. St001434The number of negative sum pairs of a signed permutation. St000264The girth of a graph, which is not a tree. St001430The number of positive entries in a signed permutation. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000260The radius of a connected graph. St000284The Plancherel distribution on integer partitions. 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. St000735The last entry on the main diagonal of a standard tableau. St000744The length of the path to the largest entry in a standard Young tableau. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St001490The number of connected components of a skew partition. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000661The number of rises of length 3 of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001141The number of occurrences of hills of size 3 in a Dyck path. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St001947The number of ties in a parking function. St001964The interval resolution global dimension of a poset. St000879The number of long braid edges in the graph of braid moves of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000456The monochromatic index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001613The binary logarithm of the size of the center of a lattice. St001617The dimension of the space of valuations of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001881The number of factors of a lattice as a Cartesian product of lattices. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000259The diameter of a connected graph. St001330The hat guessing number of a graph. St001545The second Elser number of a connected graph. St001616The number of neutral elements in a lattice. St001624The breadth of a lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001720The minimal length of a chain of small intervals in a lattice. St001754The number of tolerances of a finite lattice. St001846The number of elements which do not have a complement in the lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001427The number of descents of a signed permutation. St001625The Möbius invariant of a lattice. St000302The determinant of the distance matrix of a connected graph. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001645The pebbling number of a connected graph. St001845The number of join irreducibles minus the rank of a lattice. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St000422The energy of a graph, if it is integral. St000466The Gutman (or modified Schultz) index of a connected graph. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000464The Schultz index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000894The trace of an alternating sign matrix. St001618The cardinality of the Frattini sublattice of a lattice. St000896The number of zeros on the main diagonal of an alternating sign matrix. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001360The number of covering relations in Young's lattice below a partition. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001383The BG-rank of an integer partition. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001525The number of symmetric hooks on the diagonal of a partition. St001527The cyclic permutation representation number of an integer partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001561The value of the elementary symmetric function evaluated at 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001571The Cartan determinant of the integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001615The number of join prime elements of a lattice. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001943The sum of the squares of the hook lengths of an integer partition. St000145The Dyson rank of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000881The number of short braid edges in the graph of braid moves of a permutation. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000893The number of distinct diagonal sums of an alternating sign matrix. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001961The sum of the greatest common divisors of all pairs of parts. St000454The largest eigenvalue of a graph if it is integral. St000474Dyson's crank of a partition. St001812The biclique partition number of a graph. St000898The number of maximal entries in the last diagonal of the monotone triangle. St001030Half the number of non-boundary horizontal edges in the fully packed loop corresponding to the alternating sign matrix. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element.