#This file does not begin to find all of the equivalence classes of sets of 3 patterns of length 4 and 1 pattern of length 5. An attempt has not been made to use SipurF([4,4,4,5],4,2,10,20,n,N,x,3,2). Though as a reference point, [4,4,5,5] was not able to list all of its equivalence classes. #Begin samples of attempting to find enumeration schemes for random sets of patterns (3 length four and 1 length five). #At line 198, begin attempts with updated VatterR4S5 (lists failures and ensures K different attempts). >#VatterR4S5(3,1,100,3,2);#This line represents another run that produced nothing. VatterR4S5(3,1,100,4,2); ENUMERATION SCHEMES OF PERMUTATIONS AVOIDING, 3, PATTERNS OF LENGTH 4 AND, 1, PATTERNS OF LENGTH 5 Theorem, 1 The enumeration scheme of permutations avoiding {[1, 2, 4, 3], [2, 1, 4, 3], [3, 4, 1, 2], [5, 2, 3, 1, 4]} is {[[], {}, {}], [[1], {}, {}], [[1, 2], {}, {}], [[2, 1], {}, {}], [[1, 2, 3], {[0, 0, 1, 0]}, {1}], [[1, 3, 2], {}, {}], [[2, 1, 3], {[0, 0, 1, 0]}, {2}], [[2, 3, 1], {[0, 1, 0, 0]}, {}], [[3, 1, 2], {}, {}], [[3, 2, 1], {}, {}], [[1, 3, 2, 4], {[0, 0, 0, 1, 0], [0, 0, 1, 0, 0]}, {1}], [[1, 4, 2, 3], {[0, 0, 1, 0, 0]}, {1}], [[1, 4, 3, 2], {}, {3}], [[2, 3, 1, 4], {[0, 0, 0, 1, 0], [0, 0, 1, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 4, 1, 3], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[2, 4, 3, 1], {[0, 1, 0, 0, 0]}, {3}], [[3, 1, 2, 4], {[0, 0, 0, 1, 0], [0, 0, 1, 0, 0]}, {1}], [[3, 2, 1, 4], {[0, 0, 0, 1, 0], [0, 0, 1, 0, 0]}, {1}], [[3, 4, 1, 2], {[0, 0, 0, 0, 0]}, {1}], [[3, 4, 2, 1], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0]}, {3}], [[4, 1, 2, 3], {[0, 0, 1, 0, 0]}, {2}], [[4, 1, 3, 2], {}, {3}], [[4, 2, 1, 3], {[0, 0, 1, 0, 0]}, {3}], [[4, 2, 3, 1], {[0, 0, 0, 1, 0], [0, 1, 0, 0, 0]}, {1}], [[4, 3, 1, 2], {}, {2}], [[4, 3, 2, 1], {}, {2}]} Theorem, 2 The enumeration scheme of permutations avoiding {[1, 3, 4, 2], [2, 4, 1, 3], [3, 2, 1, 4], [3, 1, 5, 4, 2]} is {[[], {}, {}], [[1], {}, {}], [[1, 2], {}, {}], [[2, 1], {}, {}], [[1, 2, 3], {[0, 1, 0, 0]}, {1}], [[1, 3, 2], {[0, 1, 0, 1]}, {}], [[2, 1, 3], {[0, 1, 1, 0]}, {}], [[2, 3, 1], {[0, 0, 1, 0]}, {1}], [[3, 1, 2], {[0, 1, 0, 1]}, {}], [[3, 2, 1], {[0, 0, 0, 1]}, {1}], [[1, 3, 2, 4], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[1, 4, 2, 3], {[0, 0, 1, 0, 1], [0, 1, 0, 0, 0]}, {3}], [[1, 4, 3, 2], {[0, 0, 0, 0, 1], [0, 1, 0, 1, 0]}, {2}], [[2, 1, 3, 4], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0]}, {3}], [[2, 1, 4, 3], {[0, 0, 1, 0, 1], [0, 1, 0, 0, 0]}, {1}], [ [2, 4, 3, 1], {[0, 0, 0, 0, 1], [0, 0, 0, 1, 0], [0, 0, 1, 0, 0]}, {1}], [[3, 1, 2, 4], {[0, 0, 1, 1, 0], [0, 1, 0, 0, 0]}, {2}], [[3, 1, 4, 2], {[0, 0, 0, 1, 0], [0, 1, 0, 0, 1]}, {1}], [ [3, 2, 4, 1], {[0, 0, 0, 0, 1], [0, 0, 0, 1, 0], [0, 0, 1, 0, 0]}, {1}], [[4, 1, 2, 3], {[0, 0, 1, 0, 1], [0, 1, 0, 0, 0]}, {2}], [[4, 1, 3, 2], {[0, 0, 0, 0, 1], [0, 1, 0, 1, 0]}, {1}], [[4, 2, 3, 1], {[0, 0, 0, 0, 1], [0, 0, 1, 0, 0]}, {1}]} Theorem, 3 The enumeration scheme of permutations avoiding {[1, 4, 2, 3], [2, 1, 4, 3], [2, 4, 1, 3], [2, 1, 3, 4, 5]} is {[[], {}, {}], [[1], {}, {}], [[1, 2], {}, {}], [[2, 1], {}, {}], [[1, 2, 3], {}, {2}], [[1, 3, 2], {[0, 0, 1, 0]}, {}], [[2, 1, 3], {[0, 0, 0, 2], [0, 0, 1, 0]}, {1}], [[2, 3, 1], {[0, 0, 1, 0]}, {1}], [[3, 1, 2], {}, {}], [[3, 2, 1], {}, {1}], [[1, 3, 2, 4], {[0, 0, 0, 0, 2], [0, 0, 0, 1, 0], [0, 0, 1, 0, 0]}, {2}], [[1, 4, 2, 3], {[0, 0, 0, 0, 0]}, {1}], [[1, 4, 3, 2], {[0, 0, 0, 1, 0], [0, 0, 1, 0, 0]}, {2}], [[2, 4, 3, 1], {[0, 0, 0, 1, 0], [0, 0, 1, 0, 0]}, {1}], [[3, 1, 2, 4], {[0, 0, 0, 0, 2], [0, 0, 0, 1, 0]}, {1}], [[4, 1, 2, 3], {}, {3}], [[4, 1, 3, 2], {[0, 0, 1, 0, 0]}, {1}], [[4, 2, 3, 1], {[0, 0, 1, 0, 0]}, {1}]} Theorem, 4 The enumeration scheme of permutations avoiding {[2, 3, 4, 1], [3, 4, 1, 2], [4, 1, 3, 2], [3, 5, 4, 2, 1]} is {[[], {}, {}], [[1], {}, {}], [[1, 2], {}, {}], [[2, 1], {}, {}], [[1, 2, 3], {[1, 0, 0, 0]}, {1}], [[1, 3, 2], {[2, 0, 0, 0]}, {}], [[2, 1, 3], {[0, 2, 0, 0]}, {}], [[2, 3, 1], {[0, 1, 0, 0]}, {}], [[3, 1, 2], {[0, 1, 0, 0]}, {}], [[3, 2, 1], {}, {2}], [[1, 3, 2, 4], {[0, 0, 2, 0, 0], [1, 0, 0, 0, 0]}, {1}], [[1, 4, 2, 3], {[0, 0, 1, 0, 0], [1, 0, 0, 0, 0]}, {1}], [[1, 4, 3, 2], {[2, 0, 0, 0, 0]}, {3}], [[2, 1, 3, 4], {[0, 1, 0, 0, 0], [1, 0, 0, 0, 0]}, {1}], [ [2, 1, 4, 3], {[0, 2, 0, 0, 0], [1, 1, 0, 0, 0], [2, 0, 0, 0, 0]}, {2}], [ [2, 3, 1, 4], {[0, 0, 2, 0, 0], [0, 1, 0, 0, 0], [1, 0, 0, 0, 0]}, {1}], [ [2, 4, 1, 3], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0], [2, 0, 0, 0, 0]}, {1}], [[2, 4, 3, 1], {[0, 1, 0, 0, 0], [1, 0, 0, 0, 0]}, {1}], [ [3, 1, 2, 4], {[0, 0, 2, 0, 0], [0, 1, 0, 0, 0], [1, 0, 0, 0, 0]}, {2}], [ [3, 1, 4, 2], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0], [2, 0, 0, 0, 0]}, {1}], [[3, 2, 4, 1], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0]}, {1}], [[3, 4, 1, 2], {[0, 0, 0, 0, 0]}, {1}], [[3, 4, 2, 1], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0]}, {3}], [ [4, 1, 2, 3], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0], [1, 0, 0, 0, 0]}, {2}], [[4, 1, 3, 2], {[0, 0, 0, 0, 0]}, {1}], [[4, 2, 3, 1], {[0, 0, 1, 0, 0], [0, 1, 0, 0, 0]}, {4}]} The computation took, 480.595, seconds. ######################################################################################################## #After updating VatterR4S5 to list the failures: VatterR4S5(3,1,100,3,2); ENUMERATION SCHEMES OF PERMUTATIONS AVOIDING, 3, PATTERNS OF LENGTH 4 AND, 1, PATTERNS OF LENGTH 5 Theorem, 1 The enumeration scheme of permutations avoiding {[1, 4, 2, 3], [2, 1, 3, 4], [3, 1, 2, 4], [1, 3, 2, 5, 4]} is {[[], {}, {}], [[1], {}, {}], [[1, 2], {}, {}], [[2, 1], {}, {}], [[1, 2, 3], {}, {2}], [[1, 3, 2], {[0, 0, 0, 2], [0, 0, 1, 0]}, {2}], [[2, 1, 3], {[0, 0, 0, 1]}, {1}], [[2, 3, 1], {}, {2}], [[3, 1, 2], {[0, 0, 0, 1]}, {1}], [[3, 2, 1], {}, {1}]} The computation took , 36.481, seconds. SchemeFast failed for the following , 99, sets: {{[1, 2, 3, 4], [1, 2, 4, 3], [2, 1, 3, 4], [3, 4, 2, 5, 1]}, {[1, 2, 3, 4], [1, 3, 2, 4], [3, 4, 1, 2], [4, 2, 3, 1, 5]}, {[1, 2, 3, 4], [1, 3, 4, 2], [1, 4, 3, 2], [3, 4, 5, 2, 1]}, {[1, 2, 3, 4], [1, 4, 2, 3], [2, 3, 4, 1], [5, 1, 2, 4, 3]}, {[1, 2, 3, 4], [1, 4, 3, 2], [2, 3, 4, 1], [2, 1, 4, 5, 3]}, {[1, 2, 3, 4], [2, 3, 4, 1], [2, 4, 3, 1], [2, 1, 4, 3, 5]}, {[1, 2, 3, 4], [2, 3, 4, 1], [4, 1, 2, 3], [1, 4, 2, 5, 3]}, {[1, 2, 3, 4], [3, 1, 2, 4], [4, 2, 1, 3], [4, 5, 1, 3, 2]}, {[1, 2, 3, 4], [3, 2, 1, 4], [3, 4, 1, 2], [2, 1, 5, 4, 3]}, {[1, 2, 4, 3], [1, 3, 2, 4], [2, 3, 1, 4], [3, 5, 4, 1, 2]}, {[1, 2, 4, 3], [1, 3, 2, 4], [2, 4, 3, 1], [3, 1, 2, 4, 5]}, {[1, 2, 4, 3], [1, 3, 2, 4], [4, 1, 2, 3], [3, 4, 5, 1, 2]}, {[1, 2, 4, 3], [1, 3, 4, 2], [4, 2, 1, 3], [2, 4, 1, 3, 5]}, {[1, 2, 4, 3], [1, 4, 2, 3], [1, 4, 3, 2], [4, 2, 3, 5, 1]}, {[1, 2, 4, 3], [1, 4, 2, 3], [2, 3, 4, 1], [4, 5, 2, 1, 3]}, {[1, 2, 4, 3], [2, 1, 4, 3], [2, 4, 3, 1], [5, 2, 3, 1, 4]}, {[1, 2, 4, 3], [2, 1, 4, 3], [3, 1, 2, 4], [5, 2, 3, 4, 1]}, {[1, 2, 4, 3], [2, 3, 4, 1], [3, 2, 4, 1], [1, 4, 5, 3, 2]}, {[1, 2, 4, 3], [2, 3, 4, 1], [3, 2, 4, 1], [2, 1, 5, 4, 3]}, {[1, 2, 4, 3], [2, 3, 4, 1], [4, 1, 2, 3], [3, 2, 5, 4, 1]}, {[1, 2, 4, 3], [2, 3, 4, 1], [4, 1, 3, 2], [4, 2, 1, 3, 5]}, {[1, 2, 4, 3], [2, 4, 1, 3], [4, 3, 2, 1], [1, 3, 4, 2, 5]}, {[1, 2, 4, 3], [3, 1, 4, 2], [4, 3, 2, 1], [4, 2, 1, 3, 5]}, {[1, 2, 4, 3], [3, 4, 2, 1], [4, 2, 3, 1], [2, 1, 3, 4, 5]}, {[1, 3, 2, 4], [1, 3, 4, 2], [4, 2, 3, 1], [4, 2, 5, 1, 3]}, {[1, 3, 2, 4], [2, 1, 3, 4], [2, 4, 3, 1], [3, 4, 1, 2, 5]}, {[1, 3, 2, 4], [2, 4, 1, 3], [3, 1, 2, 4], [3, 5, 4, 1, 2]}, {[1, 3, 2, 4], [2, 4, 3, 1], [4, 3, 1, 2], [1, 2, 3, 5, 4]}, {[1, 3, 2, 4], [3, 1, 2, 4], [3, 4, 2, 1], [2, 1, 4, 5, 3]}, {[1, 3, 2, 4], [3, 2, 4, 1], [4, 1, 3, 2], [2, 5, 4, 1, 3]}, {[1, 3, 2, 4], [3, 4, 2, 1], [4, 3, 1, 2], [3, 2, 1, 5, 4]}, {[1, 3, 2, 4], [4, 2, 1, 3], [4, 3, 2, 1], [5, 1, 4, 2, 3]}, {[1, 3, 4, 2], [1, 4, 2, 3], [4, 2, 3, 1], [5, 4, 1, 3, 2]}, {[1, 3, 4, 2], [2, 1, 3, 4], [3, 2, 1, 4], [5, 1, 2, 4, 3]}, {[1, 3, 4, 2], [2, 1, 4, 3], [3, 4, 2, 1], [4, 2, 5, 1, 3]}, {[1, 3, 4, 2], [2, 1, 4, 3], [4, 3, 2, 1], [3, 4, 5, 2, 1]}, {[1, 3, 4, 2], [2, 3, 1, 4], [3, 2, 4, 1], [2, 1, 3, 5, 4]}, {[1, 3, 4, 2], [2, 3, 1, 4], [4, 2, 3, 1], [2, 1, 3, 4, 5]}, {[1, 3, 4, 2], [3, 1, 4, 2], [4, 1, 3, 2], [3, 2, 5, 4, 1]}, {[1, 3, 4, 2], [4, 1, 2, 3], [4, 3, 2, 1], [1, 5, 4, 2, 3]}, {[1, 3, 4, 2], [4, 2, 1, 3], [4, 3, 1, 2], [1, 2, 5, 4, 3]}, {[1, 4, 2, 3], [1, 4, 3, 2], [2, 3, 1, 4], [3, 4, 5, 2, 1]}, {[1, 4, 2, 3], [1, 4, 3, 2], [4, 1, 2, 3], [4, 5, 3, 2, 1]}, {[1, 4, 2, 3], [2, 1, 3, 4], [2, 3, 1, 4], [4, 3, 5, 1, 2]}, {[1, 4, 2, 3], [2, 1, 3, 4], [3, 4, 1, 2], [3, 1, 2, 5, 4]}, {[1, 4, 2, 3], [2, 1, 3, 4], [3, 4, 2, 1], [3, 5, 1, 4, 2]}, {[1, 4, 2, 3], [2, 3, 1, 4], [3, 1, 2, 4], [5, 4, 1, 2, 3]}, {[1, 4, 2, 3], [2, 3, 1, 4], [3, 4, 2, 1], [4, 5, 1, 2, 3]}, {[1, 4, 2, 3], [3, 1, 4, 2], [3, 2, 1, 4], [2, 5, 4, 1, 3]}, {[1, 4, 2, 3], [3, 1, 4, 2], [4, 3, 2, 1], [3, 4, 1, 2, 5]}, {[1, 4, 3, 2], [2, 1, 4, 3], [4, 2, 1, 3], [2, 4, 5, 1, 3]}, {[1, 4, 3, 2], [2, 1, 4, 3], [4, 2, 3, 1], [4, 5, 3, 1, 2]}, {[1, 4, 3, 2], [2, 3, 1, 4], [3, 2, 4, 1], [4, 5, 3, 1, 2]}, {[1, 4, 3, 2], [2, 4, 3, 1], [3, 2, 4, 1], [1, 4, 5, 2, 3]}, {[1, 4, 3, 2], [3, 1, 4, 2], [4, 2, 3, 1], [1, 4, 5, 2, 3]}, {[1, 4, 3, 2], [3, 2, 1, 4], [4, 1, 2, 3], [4, 3, 5, 1, 2]}, {[1, 4, 3, 2], [3, 2, 1, 4], [4, 1, 3, 2], [5, 3, 4, 1, 2]}, {[1, 4, 3, 2], [3, 2, 4, 1], [4, 1, 2, 3], [3, 4, 5, 2, 1]}, {[2, 1, 3, 4], [2, 1, 4, 3], [3, 2, 1, 4], [3, 5, 4, 1, 2]}, {[2, 1, 3, 4], [2, 1, 4, 3], [3, 2, 4, 1], [2, 3, 5, 4, 1]}, {[2, 1, 3, 4], [2, 3, 1, 4], [2, 4, 3, 1], [1, 4, 2, 5, 3]}, {[2, 1, 3, 4], [2, 3, 1, 4], [3, 1, 2, 4], [4, 3, 1, 5, 2]}, {[2, 1, 3, 4], [2, 3, 4, 1], [2, 4, 3, 1], [5, 1, 2, 3, 4]}, {[2, 1, 3, 4], [3, 2, 1, 4], [4, 2, 3, 1], [2, 3, 4, 1, 5]}, {[2, 1, 4, 3], [2, 4, 1, 3], [3, 2, 4, 1], [3, 1, 4, 2, 5]}, {[2, 1, 4, 3], [2, 4, 1, 3], [3, 2, 4, 1], [5, 3, 1, 2, 4]}, {[2, 1, 4, 3], [2, 4, 1, 3], [3, 4, 2, 1], [1, 5, 3, 2, 4]}, {[2, 1, 4, 3], [4, 1, 3, 2], [4, 2, 3, 1], [3, 5, 2, 1, 4]}, {[2, 1, 4, 3], [4, 1, 3, 2], [4, 3, 2, 1], [5, 2, 3, 4, 1]}, {[2, 3, 1, 4], [2, 4, 1, 3], [4, 1, 2, 3], [4, 3, 5, 1, 2]}, {[2, 3, 1, 4], [3, 1, 2, 4], [4, 1, 3, 2], [5, 3, 2, 1, 4]}, {[2, 3, 1, 4], [3, 2, 4, 1], [3, 4, 2, 1], [5, 4, 1, 3, 2]}, {[2, 3, 4, 1], [2, 4, 1, 3], [2, 4, 3, 1], [4, 5, 1, 2, 3]}, {[2, 3, 4, 1], [2, 4, 1, 3], [4, 2, 3, 1], [4, 3, 2, 5, 1]}, {[2, 3, 4, 1], [3, 1, 2, 4], [4, 1, 3, 2], [4, 3, 2, 1, 5]}, {[2, 3, 4, 1], [3, 1, 4, 2], [3, 2, 1, 4], [4, 5, 3, 2, 1]}, {[2, 3, 4, 1], [3, 2, 1, 4], [3, 2, 4, 1], [1, 4, 2, 3, 5]}, {[2, 3, 4, 1], [3, 2, 4, 1], [4, 1, 3, 2], [3, 2, 1, 4, 5]}, {[2, 3, 4, 1], [3, 4, 1, 2], [4, 3, 2, 1], [5, 1, 3, 4, 2]}, {[2, 3, 4, 1], [3, 4, 2, 1], [4, 3, 1, 2], [4, 1, 2, 3, 5]}, {[2, 3, 4, 1], [4, 1, 3, 2], [4, 2, 1, 3], [3, 2, 5, 4, 1]}, {[2, 4, 1, 3], [2, 4, 3, 1], [3, 1, 2, 4], [5, 4, 3, 1, 2]}, {[2, 4, 1, 3], [2, 4, 3, 1], [4, 2, 3, 1], [3, 4, 2, 1, 5]}, {[2, 4, 1, 3], [3, 1, 4, 2], [3, 4, 2, 1], [5, 4, 1, 3, 2]}, {[2, 4, 1, 3], [3, 1, 4, 2], [4, 1, 2, 3], [4, 5, 1, 3, 2]}, {[2, 4, 1, 3], [3, 2, 4, 1], [4, 1, 3, 2], [2, 3, 1, 4, 5]}, {[2, 4, 1, 3], [4, 1, 2, 3], [4, 3, 2, 1], [3, 2, 4, 1, 5]}, {[2, 4, 3, 1], [3, 1, 2, 4], [4, 3, 1, 2], [4, 5, 2, 1, 3]}, {[2, 4, 3, 1], [3, 2, 4, 1], [4, 1, 3, 2], [1, 2, 5, 3, 4]}, {[2, 4, 3, 1], [3, 2, 4, 1], [4, 2, 1, 3], [4, 1, 2, 5, 3]}, {[3, 1, 2, 4], [3, 2, 1, 4], [4, 2, 3, 1], [3, 1, 5, 4, 2]}, {[3, 1, 2, 4], [3, 2, 4, 1], [4, 3, 1, 2], [1, 3, 2, 5, 4]}, {[3, 1, 2, 4], [3, 4, 1, 2], [3, 4, 2, 1], [5, 4, 2, 3, 1]}, {[3, 1, 4, 2], [3, 2, 1, 4], [4, 2, 3, 1], [1, 2, 4, 5, 3]}, {[3, 1, 4, 2], [4, 1, 2, 3], [4, 3, 2, 1], [1, 3, 4, 2, 5]}, {[3, 1, 4, 2], [4, 2, 3, 1], [4, 3, 1, 2], [1, 3, 2, 5, 4]}, {[3, 2, 1, 4], [4, 2, 3, 1], [4, 3, 2, 1], [5, 4, 1, 2, 3]}, {[3, 2, 4, 1], [4, 1, 2, 3], [4, 2, 1, 3], [1, 4, 5, 2, 3]}, {[3, 4, 1, 2], [3, 4, 2, 1], [4, 1, 3, 2], [5, 3, 2, 4, 1]}}