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Statistic identifier: St000635
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Collection: Posets
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Description: The number of strictly order preserving maps of a poset into itself.
A map $f$ is strictly order preserving if $a < b$ implies $f(a) < f(b)$.
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References: [1] Alexandersson, P. Whether a total order set of size $n$ has the fewest endomorphisms among posets of size $n$ [[MathOverflow:252913]]
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Code:
def is_strictly_order_preserving(f, P):
return all([f(a),f(b)] in P.cover_relations() for (a, b) in P.cover_relations())
def statistic(P):
P = P.relabel()
r = P.cardinality()
S = cartesian_product([range(r)]*r)
return len([pi for pi in S if is_strictly_order_preserving(lambda i: pi[i], P)])
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Statistic values:
([],2) => 4
([(0,1)],2) => 1
([],3) => 27
([(1,2)],3) => 3
([(0,1),(0,2)],3) => 4
([(0,2),(2,1)],3) => 1
([(0,2),(1,2)],3) => 4
([],4) => 256
([(2,3)],4) => 16
([(1,2),(1,3)],4) => 16
([(0,1),(0,2),(0,3)],4) => 27
([(0,2),(0,3),(3,1)],4) => 2
([(0,1),(0,2),(1,3),(2,3)],4) => 4
([(1,2),(2,3)],4) => 4
([(0,3),(3,1),(3,2)],4) => 4
([(1,3),(2,3)],4) => 16
([(0,3),(1,3),(3,2)],4) => 4
([(0,3),(1,3),(2,3)],4) => 27
([(0,3),(1,2)],4) => 4
([(0,3),(1,2),(1,3)],4) => 8
([(0,2),(0,3),(1,2),(1,3)],4) => 16
([(0,3),(2,1),(3,2)],4) => 1
([(0,3),(1,2),(2,3)],4) => 2
([],5) => 3125
([(3,4)],5) => 125
([(2,3),(2,4)],5) => 100
([(1,2),(1,3),(1,4)],5) => 135
([(0,1),(0,2),(0,3),(0,4)],5) => 256
([(0,2),(0,3),(0,4),(4,1)],5) => 9
([(0,1),(0,2),(0,3),(2,4),(3,4)],5) => 12
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 27
([(1,3),(1,4),(4,2)],5) => 10
([(0,3),(0,4),(4,1),(4,2)],5) => 8
([(1,2),(1,3),(2,4),(3,4)],5) => 20
([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 4
([(0,3),(0,4),(3,2),(4,1)],5) => 4
([(0,2),(0,3),(2,4),(3,1),(3,4)],5) => 8
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => 16
([(2,3),(3,4)],5) => 25
([(1,4),(4,2),(4,3)],5) => 20
([(0,4),(4,1),(4,2),(4,3)],5) => 27
([(2,4),(3,4)],5) => 100
([(1,4),(2,4),(4,3)],5) => 20
([(0,4),(1,4),(4,2),(4,3)],5) => 16
([(1,4),(2,4),(3,4)],5) => 135
([(0,4),(1,4),(2,4),(4,3)],5) => 27
([(0,4),(1,4),(2,4),(3,4)],5) => 256
([(0,4),(1,4),(2,3)],5) => 15
([(0,4),(1,3),(2,3),(2,4)],5) => 24
([(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 55
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 108
([(0,4),(1,4),(2,3),(4,2)],5) => 4
([(0,4),(1,3),(2,3),(3,4)],5) => 8
([(0,4),(1,4),(2,3),(2,4)],5) => 38
([(0,4),(1,4),(2,3),(3,4)],5) => 9
([(1,4),(2,3)],5) => 20
([(1,4),(2,3),(2,4)],5) => 40
([(0,4),(1,2),(1,4),(2,3)],5) => 3
([(0,3),(1,2),(1,3),(2,4),(3,4)],5) => 8
([(1,3),(1,4),(2,3),(2,4)],5) => 80
([(0,3),(0,4),(1,3),(1,4),(4,2)],5) => 8
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => 16
([(0,4),(1,2),(1,4),(4,3)],5) => 6
([(0,4),(1,2),(1,3)],5) => 15
([(0,4),(1,2),(1,3),(1,4)],5) => 38
([(0,2),(0,4),(3,1),(4,3)],5) => 2
([(0,4),(1,2),(1,3),(3,4)],5) => 4
([(0,2),(0,3),(1,4),(2,4),(3,1)],5) => 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 12
([(0,3),(0,4),(1,2),(1,4)],5) => 24
([(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 55
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 108
([(0,3),(0,4),(1,2),(1,3),(2,4)],5) => 1
([(0,3),(1,2),(1,4),(3,4)],5) => 3
([(0,3),(0,4),(1,2),(2,3),(2,4)],5) => 8
([(1,4),(3,2),(4,3)],5) => 5
([(0,3),(3,4),(4,1),(4,2)],5) => 4
([(1,4),(2,3),(3,4)],5) => 10
([(0,4),(1,2),(2,4),(4,3)],5) => 2
([(0,3),(1,4),(4,2)],5) => 3
([(0,4),(3,2),(4,1),(4,3)],5) => 2
([(0,4),(1,2),(2,3),(2,4)],5) => 6
([(0,4),(2,3),(3,1),(4,2)],5) => 1
([(0,3),(1,2),(2,4),(3,4)],5) => 4
([(0,4),(1,2),(2,3),(3,4)],5) => 2
([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 4
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Created: Oct 25, 2016 at 12:01 by Martin Rubey
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Last Updated: Nov 13, 2022 at 11:37 by Martin Rubey