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object_name => value

    E.g.

    [1,2,3] => 1
    [1,3,2] => 1
    [2,1,3] => 2
    [2,3,1] => 2

Detailed description of the statistic.

    E.g.

    The major index of a permutation.

    The '''major index''' of a permutation $\sigma$ is defined as
    $$\sum_{i \in Des(\sigma)} i$$
    where $Des(\sigma)$ is the set of all indices $i$ such that $\sigma(i) > \sigma(i+1)$.

References to literature describing the statistic, if applicable.
Preferably, references are given by one or multiple identifiers.
Author and title are retrieved automatically, if possible.

    E.g.

    [1] [[mathscinet:3113227]] [[arxiv:1304.3866]] [[doi:10.1090/noti1029]]
    [2] [[zbmath:1218.68204]] [[arXiv:math/0702086]]
    [3] [[mathoverflow:132338]]
    [4] [[oeis:A000108]]
    [5] [[wikipedia:On-Line_Encyclopedia_of_Integer_Sequences]]
    [6] free description [[http://something]]

# User provided code
def statistic(x):
    # Your code
    return value_at_x
            

Code of the statistic, if applicable.

    E.g.

    def statistic(x):
        # Your code
        return value_at_x

# edit this # WARNING: takes a long time even for permutations of 5. def defect_set(sigma, w): """ INPUT: - w is a reduced word for a permutation - sigma is the mask, given as a binary word of length l(w) OUTPUT: - the defect set, that is the set of positions j such that l(w^{\sigma[j-1]} w_j) < l(w^{\sigma[j-1]}) """ res = [] S = Permutations(max(w)+1) w_S = [S.simple_reflection(e) for e in w] w_up_to_j = S([]) for j in range(len(w)-1): if sigma[j] == 1: w_up_to_j = w_up_to_j.left_action_product(w_S[j]) if (w_up_to_j.left_action_product(w_S[j+1])).length() < w_up_to_j.length(): res += [j+2] return res def statistic(pi): """ sage: statistic(Permutation([4,3,5,2,1])) 3 sage: statistic(Permutation([5,4,2,1,3])) 3 """ k = pi.length() if not k: return 0 masks = cartesian_product([[0,1]]*k) return max([len(defect_set(m, w)) for m in masks for w in pi.reduced_words()]) # DON'T edit this parent_initializer = Permutations element_repr = repr levels = [2, 3, 4, 5] for level in levels: for elt in parent_initializer(level): print('%s => %s' % (element_repr(elt), statistic(elt)))

object_name => value E.g. [] => 0 [1] => 0 [1,2] => 0 [2,1] => 0 [1,2,3] => 0

# edit this @cached_function def P(k): return posets.IntegerPartitionsDominanceOrder(k) def statistic(pi): Q = P(pi.size()) return len(Q.lower_covers(Q(pi))) # DON'T edit this parent_initializer = Partitions element_repr = repr levels = [2, 3, 4, 5, 6, 7, 8, 9, 10] for level in levels: for elt in parent_initializer(level): print('%s => %s' % (element_repr(elt), statistic(elt)))

object_name => value E.g. [] => 0 [1] => 0 [2] => 1 [1,1] => 0 [3] => 1

# edit this # depends on https://trac.sagemath.org/ticket/16110 def statistic(D): return ParallelogramPolyomino.from_dyck_word(D).area() # DON'T edit this parent_initializer = DyckWords element_repr = repr levels = [2, 3, 4, 5, 6] for level in levels: for elt in parent_initializer(level): print('%s => %s' % (element_repr(elt), statistic(elt)))

object_name => value E.g. [] => 0 [1,0] => 1 [1,0,1,0] => 2 [1,1,0,0] => 2 [1,0,1,0,1,0] => 3

# edit this def avoids(w, pattern): l = [len(p) for p in pattern] l_p = len(pattern) n = len(w) A = sorted(list(set([a for p in pattern for a in p]))) pattern = [[A.index(a) for a in p] for p in pattern] k = 1+max(max(p) for p in pattern) def is_match(s): assignment = [None]*k for i in range(l_p): for j in range(len(pattern[i])): l = w[s[i]+j] p = pattern[i][j] m = assignment[p] if m is None: assignment[p] = l elif m != l: return False return all(assignment[i] < assignment[i+1] for i in range(k-1)) for s in Subsets(list(range(len(w))), l_p)._fast_iterator(): if all(s[i]+l[i] <= s[i+1] for i in range(l_p-1)) and s[-1]+l[-1] <= n: if is_match(s): return False return True def statistic(la): mset = [i for i, p in enumerate(la) for _ in range(p)] return len([w for w in Permutations(mset)if avoids(w, [[1,3,2]])]) # DON'T edit this parent_initializer = Compositions element_repr = repr levels = [2, 3, 4, 5, 6, 7] for level in levels: for elt in parent_initializer(level): print('%s => %s' % (element_repr(elt), statistic(elt)))

object_name => value E.g. [] => 1 [1] => 1 [1,1] => 2 [2] => 1 [1,1,1] => 5

# edit this def diagonal_index_of_partition(la): return sum((j - i) for (i, j) in la.cells()) def binomial_shifted_diagonal_index_of_partition(partition): return binomial(partition.size() + 1, 2) + diagonal_index_of_partition(partition) def is_desarrangement_tableau(t): if t.size() == 0: return True descents = t.standard_descents() ascents = [i for i in t.entries() if i not in descents] return min(ascents) % 2 == 0 def r2r_statistic_on_standard_tableaux(t): # remove 1 and rectify until we get a desarrangement tableau s = copy(t) while not is_desarrangement_tableau(s): s = SkewTableau([[(i-1 if i > 1 else None) for i in row] for row in s]) s = StandardTableau(s.rectify()) b_t = binomial_shifted_diagonal_index_of_partition(t.shape()) b_s = binomial_shifted_diagonal_index_of_partition(s.shape()) return b_t - b_s def statistic(t): return r2r_statistic_on_standard_tableaux(t) # DON'T edit this parent_initializer = StandardTableaux element_repr = repr levels = [2, 3, 4, 5, 6] for level in levels: for elt in parent_initializer(level): print('%s => %s' % (element_repr(elt), statistic(elt)))

object_name => value E.g. [[1,2,3,4,6,7,8],[5,9]] => 39 [[1,2,4,5,6,8,9],[3],[7]] => 15 [[1,2,5,6,8,9],[3,4,7]] => 10 [[1,3,4,6,8,9],[2,5,7]] => 0 [[1,2,3,6,8,9],[4,5,7]] => 23

# edit this def Klazar_occurrences(pi, pattern): """ Return the number of occurrences of pattern in pi. EXAMPLES: From Mansour 2013, page 86, Example 3.24 The set partition π = 135/26/4 contains four occurrences of the pattern 12/3, namely, 13/6, 13/4, 15/6, and 35/6. On the other hand, the set partition π avoids the pattern 12/34. sage: Klazar_occurrences([[1,3,5],[2,6],[4]], [[1,2],[3]]) 4 sage: Klazar_occurrences([[1,3,5],[2,6],[4]], [[1,2],[3,4]]) 0 """ pi = SetPartition(pi).standardization() pattern = SetPartition(pattern).standardization() count = 0 for s in Subsets(range(1,1+pi.size()), pattern.size()): if pi.restriction(s).standardization() == pattern: count += 1 return count def statistic(pi): return Klazar_occurrences(pi, [[1,2,3]]) # DON'T edit this parent_initializer = SetPartitions element_repr = lambda X: str(sorted([sorted(s) for s in X])).replace("[","{").replace("]","}") levels = [2, 3, 4, 5] for level in levels: for elt in parent_initializer(level): print('%s => %s' % (element_repr(elt), statistic(elt)))

object_name => value E.g. {{1,2}} => 0 {{1},{2}} => 0 {{1,2,3}} => 1 {{1},{2,3}} => 0 {{1,3},{2}} => 0

# edit this # BinaryTree -> Bool def is_leaf(b): return b == BinaryTree('.') # BinaryTree -> Int def get_external_node(b, flag): count = 0; if is_leaf(b): return 0 else: if (flag): # True counts the left side if not is_leaf(b[0]): return 1 + get_external_node(b[0], True) else: # False counts the right side if not is_leaf(b[1]): return 1 + get_external_node(b[1], False) return count # BinaryTree -> Int def external_node_count(b): return 1 + get_external_node(b, True) + get_external_node(b, False) def size(b): if not b: return 1 else: return size(b[0])+size(b[1]) def statistic(b): return size(b)-1-external_node_count(b) # DON'T edit this parent_initializer = BinaryTrees element_repr = repr levels = [2, 3, 4, 5, 6] for level in levels: for elt in parent_initializer(level): print('%s => %s' % (element_repr(elt), statistic(elt)))

object_name => value E.g. [.,.] => 0 [.,[.,.]] => 0 [[.,.],.] => 0 [.,[.,[.,.]]] => 0 [.,[[.,.],.]] => 1

# edit this @cached_function def statistic(w): def children(m): for (a,b),(c,d) in m.crossings(): m_new = list(m) m_new.remove((a,b)) m_new.remove((c,d)) A, B = min(a,b), max(a,b) C, D = min(c,d), max(c,d) if C < A: (A,B),(C,D) = (C,D),(A,B) yield PerfectMatching(m_new + [(A,C), (B,D)]) yield PerfectMatching(m_new + [(A,D), (B,C)]) w = PerfectMatching(sorted([sorted(a) for a in w])) l = [statistic(v) for v in children(w)] if len(l) == 0: return 0 else: return 1+min(l) # DON'T edit this parent_initializer = PerfectMatchings element_repr = lambda m: str(sorted(tuple(sorted(b)) for b in m)) levels = [2, 4, 6, 8] for level in levels: for elt in parent_initializer(level): print('%s => %s' % (element_repr(elt), statistic(elt)))

object_name => value E.g. [(1,2)] => 0 [(1,2),(3,4)] => 0 [(1,3),(2,4)] => 1 [(1,4),(2,3)] => 0 [(1,2),(3,4),(5,6)] => 0

# edit this def statistic(C): return C.length() # DON'T edit this parent_initializer = lambda x: Cores(x[1],x[0]) element_repr = lambda X: "( "+X._repr_()+", "+str(X.k())+" )" levels = [(2, 3), (3, 3), (4, 3), (5, 3), (6, 3), (2, 4), (3, 4), (4, 4), (5, 4), (6, 4), (2, 5), (3, 5), (4, 5), (5, 5), (6, 5), (2, 6), (3, 6), (4, 6), (5, 6), (6, 6), (7, 6)] for level in levels: for elt in parent_initializer(level): print('%s => %s' % (element_repr(elt), statistic(elt)))

object_name => value E.g. ([2,1],2) => 2 ([2],3) => 2 ([1,1],3) => 2 ([2],4) => 2 ([1,1],4) => 2

# edit this # code by Jang Soo Kim def delete_elements(P,A): "Return the poset obtained from P by deleting the elements in A" elms = [x for x in P.list() if x not in A] rels = [R for R in P.relations() if R[0] not in A and R[1] not in A] return Poset([elms,rels], cover_relations=False) def add_bottom(P): "Return the poset obtained from P by adding the minimum element 0." elms = ["zero_hat"] + P.list() rels = [["zero_hat",x] for x in P.list()] + P.relations() return Poset([elms,rels], cover_relations=False) def Poin(P,t,omega=0): """ Input: P: a poset or a list of numbers [a,b,c,...]. If P = [a,b,c,...], then P is set to be the cell poset of the partition [a,b,c,...]. t: a variable omega: a labeling of P which is given by default to the first lin ext of P. Output: The Poincare polynomial of the poset P. The bijection from linear extensions to P-transverse permutations is used. """ if type(P) == list: P = Partition(P).cell_poset().dual() if omega == 0: omega = P.linear_extensions()[0] poly = 0 for L in P.linear_extensions(): poly += t^(len(P)-len(get_P_transverse_permutation(P,omega,L))) return poly def get_P_transverse_permutation(P,omega,L): """ Input: P: a poset omega: a labeling of P L: a linear extension of P Output: The P-transverse permutation corresponding to L with respect to omega. """ Q = add_bottom(P) pi = [] while len(L)>0: N = next_level(Q,P,omega,L) pi = pi + N[-1] [Q, P, omega, L] = N[:-1] return pi def next_level(Q,P,omega,L): """ INPUT: Q,P,omega,L Q: poset containing P omega: labeling L: a linear extension of P OUTPUT: [Q1, P1, omega, L1, pi] Q1 = P P1 = P - minimal blocks omega = omega L1 = L restricted to P1 pi = P-transverse permutation of the minimal blocks in P. """ L1 = L pi = [] C = [] # the list of current level elements in P D = [] # the list of current level elements in P that are not minimal in Q B = [] for x in L: if x not in P.minimal_elements(): # If x is not a minimal element in P this level is done. break C.append(x) if x not in Q.minimal_elements(): D.append(x) if omega.index(x) == min([omega.index(d) for d in D]): if len(B)>0: pi.append(B) B = [x] else: B.append(x) pi.append(B) Q1 = P P1 = delete_elements(P,C) L1 = L[len(C):] return [Q1, P1, omega, L1, pi] statistic = lambda P: Poin(P, -1) # DON'T edit this def parent_initializer(n): class MyPosets(): def __iter__(self): for P in Posets(n): yield P def cardinality(self): l = [1, 1, 2, 5, 16, 63, 318, 2045, 16999, 183231, 2567284, 46749427, 1104891746, 33823827452, 1338193159771, 68275077901156, 4483130665195087] if n < len(l): return l[n] else: return Infinity return MyPosets() element_repr = lambda X: str( ( sorted(X._hasse_diagram.canonical_label().cover_relations()), len(X._hasse_diagram.vertices(sort=False)) ) ) levels = [2, 3, 4, 5] for level in levels: for elt in parent_initializer(level): print('%s => %s' % (element_repr(elt), statistic(elt)))

object_name => value E.g. ([],1) => 1 ([],2) => 0 ([(0,1)],2) => 1 ([],3) => 0 ([(1,2)],3) => -1

# edit this def ASM_X_ray(A): P = A.to_matrix() n = P.nrows() return tuple(sum(P[k-1-j][j] for j in range(max(0, k-n), min(k,n))) for k in range(1,2*n)) @cached_function def ASM_X_rays(n): return sorted(ASM_X_ray(A) for A in AlternatingSignMatrices(n)) def statistic(A): return ASM_X_rays(A.to_matrix().nrows()).count(ASM_X_ray(A)) # DON'T edit this parent_initializer = AlternatingSignMatrices element_repr = lambda X: str([list(row) for row in X.to_matrix()]) levels = [2, 3, 4] for level in levels: for elt in parent_initializer(level): print('%s => %s' % (element_repr(elt), statistic(elt)))

object_name => value E.g. [[1]] => 1 [[1,0],[0,1]] => 1 [[0,1],[1,0]] => 1 [[1,0,0],[0,1,0],[0,0,1]] => 1 [[0,1,0],[1,0,0],[0,0,1]] => 1

# edit this def statistic(G): return G[0][0]-G[0][-1] # DON'T edit this def parent_initializer(x): class MyGTP(): def __iter__(self): for la in Partitions(x[1], max_length=x[0]): top_row = la + [0]*(x[0]-len(la)) for P in GelfandTsetlinPatterns(n=x[0], top_row=top_row): yield P def cardinality(self): return SemistandardTableaux(size=x[1], max_entry=x[0]).cardinality() return MyGTP() element_repr = repr levels = [(2, 1), (2, 2), (3, 1), (2, 3), (3, 2), (4, 1), (2, 4), (3, 3), (4, 2), (5, 1), (2, 5), (3, 4), (4, 3), (5, 2), (6, 1)] for level in levels: for elt in parent_initializer(level): print('%s => %s' % (element_repr(elt), statistic(elt)))

object_name => value E.g. [[1,0],[0]] => 1 [[1,0],[1]] => 1 [[2,0],[0]] => 2 [[2,0],[1]] => 2 [[2,0],[2]] => 2

# edit this def statistic(T): S = [[e for e in row] for row in T] count = 0 for i, row in enumerate(S): for j, e in enumerate(row): S[i][j] = e+1 try: _ = SemistandardTableau(S) except StandardError: count += 1 S[i][j] = e return count # DON'T edit this def parent_initializer(x): class MySemistandardTableaux(): def __iter__(self): for T in SemistandardTableaux(size=x[0], max_entry=x[1]): if max(T.entries()) == x[1]: yield T def cardinality(self): c1 = SemistandardTableaux(size=x[0], max_entry=x[1]).cardinality() if x[1] <= 1: return c1 c2 = SemistandardTableaux(size=x[0], max_entry=x[1]-1).cardinality() return c1 - c2 return MySemistandardTableaux() element_repr = repr levels = [(2, 2), (2, 3), (3, 2), (2, 4), (3, 3), (4, 2), (2, 5), (3, 4), (4, 3), (5, 2), (2, 6), (3, 5), (4, 4), (5, 3), (6, 2)] for level in levels: for elt in parent_initializer(level): print('%s => %s' % (element_repr(elt), statistic(elt)))

object_name => value E.g. [[1,2]] => 0 [[2,2]] => 1 [[1],[2]] => 1 [[1,3]] => 0 [[2,3]] => 0

# edit this """ On the dimension of a graph, Paul Erdös, Frank Harary and William T. Tutte """ dimensions = dict() N = 7 for n in range(1,N+1): # C_n if n > 3: G = graphs.CycleGraph(n) dimensions[G.canonical_label().copy(immutable=True)] = 2 # P_n if n > 1: G = graphs.PathGraph(n) dimensions[G.canonical_label().copy(immutable=True)] = 1 # K_n G = graphs.CompleteGraph(n) dimensions[G.canonical_label().copy(immutable=True)] = n-1 # K_n-e if n > 2: G.delete_edge(G.edges()[0]) dimensions[G.canonical_label().copy(immutable=True)] = n-2 # K_{2,n} G = graphs.CompleteBipartiteGraph(2, n) if n >= 3: dimensions[G.canonical_label().copy(immutable=True)] = 3 elif n == 2: dimensions[G.canonical_label().copy(immutable=True)] = 2 # K_{3,n} G = graphs.CompleteBipartiteGraph(3, n) if n >= 3: dimensions[G.canonical_label().copy(immutable=True)] = 4 # Wheel G = graphs.WheelGraph(n) # n vertices if n in [5, 6]: dimensions[G.canonical_label().copy(immutable=True)] = 3 elif n == 7: dimensions[G.canonical_label().copy(immutable=True)] = 2 elif n > 7: dimensions[G.canonical_label().copy(immutable=True)] = 3 G = graphs.CubeGraph(n) # 2^n vertices if n > 1: dimensions[G.canonical_label().copy(immutable=True)] = 2 def statistic(G): global dimensions # bridges do not change the dimension, unless we have a forest G = G.copy(immutable=False) bridges = list(G.bridges()) if G.num_edges() == len(bridges): if G.num_edges() == 0: return 0 if max(G.degree()) <= 2: return 1 return 2 G.delete_edges(bridges) l = G.connected_components_subgraphs() if len(l) > 1: statistics = [statistic(H) for H in l] if not any(v is None for v in statistics): return max(statistics) l = G.blocks_and_cut_vertices()[0] if len(l) > 1: statistics = [statistic(G.subgraph(B)) for B in l] if not any(v is None for v in statistics): return max(statistics) G = G.canonical_label().copy(immutable=True) if G in dimensions: return dimensions[G] n = G.num_verts() if G.is_clique(): return n-1 # DON'T edit this def parent_initializer(n): class MyGraphs(): def __iter__(self): for G in graphs(n): G._immutable = True yield G def cardinality(self): l = [1, 1, 2, 4, 11, 34, 156, 1044, 12346, 274668] if n < len(l): return l[n] else: return Infinity return MyGraphs() element_repr = lambda X: str((sorted(X.canonical_label().edges(sort=False, labels=False)), X.num_verts())) levels = [1, 2, 3, 4, 5, 6] for level in levels: for elt in parent_initializer(level): print('%s => %s' % (element_repr(elt), statistic(elt)))

object_name => value E.g. ([],0) => 0 ([],1) => 0 ([],2) => 0 ([(0,1)],2) => 1 ([],3) => 0

# edit this @cached_function def Ordered_to_Rooted(n): return [RootedTree(S) for S in OrderedTrees(n)] def statistic(tree): return Ordered_to_Rooted(tree.node_number()).count(RootedTree(tree)) # DON'T edit this parent_initializer = OrderedTrees element_repr = repr levels = [2, 3, 4, 5, 6, 7] for level in levels: for elt in parent_initializer(level): print('%s => %s' % (element_repr(elt), statistic(elt)))

object_name => value E.g. [[]] => 1 [[],[]] => 1 [[[]]] => 1 [[],[],[]] => 1 [[],[[]]] => 2

# edit this @cached_function def statistic(ct): return sum(1 for w in WeylGroup(ct) if is_proper(ct, w)) def is_proper(ct, w): ct = CartanType(ct) n = ct.rank() l = w.length() d = len(w.descents(side="left")) if ct.type() == "A": return l <= n + binomial(d + 1, 2) if ct.type() in ["B", "C"]: return l <= n + d^2 if ct.type() == "D": if d > 3: return l <= n + d^2 - d return l <= binomial(d + 1, 2) if ct.type() == "E": m = [0, 1, 3, 6, 12, 20, 36, 63, 120] return l <= n + m[d] if ct.type() == "F": m = [0, 1, 4, 9, 24] return l <= n + m[d] if ct.type() == "G": # = I_2(6) if d == 2: return l <= 6 + n if d < 2: return l <= 2 + d # DON'T edit this def parent_initializer(n): cartan_types = [ ['A',n] ] if n >= 2: cartan_types += [ ['B',n] ] if n >= 3: cartan_types += [ ['C',n] ] if n >= 4: cartan_types += [ ['D',n] ] if n in [6,7,8]: cartan_types += [ ['E',n] ] if n == 4: cartan_types += [ ['F',n] ] if n == 2: cartan_types += [ ['G',n] ] return [ CartanType(cartan_type) for cartan_type in cartan_types ] element_repr = repr levels = [1, 2, 3, 4, 5, 6, 7, 8] for level in levels: for elt in parent_initializer(level): print('%s => %s' % (element_repr(elt), statistic(elt)))

object_name => value E.g. ['A',1] => 2 ['A',2] => 6 ['B',2] => 8 ['G',2] => 8 ['A',3] => 24

# edit this def statistic(P): """ sage: R. = PolynomialRing(ZZ) sage: n=3; sum(x^statistic(P)*y^(binomial(len(P)+1,2)-sum(P)) for P in ParkingFunctions(n)) x^3 + y^3 + 3*x^2 + 4*x*y + 3*y^2 + 2*x + 2*y sage: G = graphs.CompleteGraph(n+1) sage: G.tutte_polynomial() x^3 + y^3 + 3*x^2 + 4*x*y + 3*y^2 + 2*x + 2*y """ return len([1 for i, j in enumerate(P) if ((i == 0 or max(P[:i]) < j) and len([1 for k in P if k < j]) == j-1 and len([1 for k in P if k > j]) == len(P)-j)]) # DON'T edit this parent_initializer = ParkingFunctions element_repr = repr levels = [1, 2, 3, 4] for level in levels: for elt in parent_initializer(level): print('%s => %s' % (element_repr(elt), statistic(elt)))

object_name => value E.g. [1] => 1 [1,1] => 0 [1,2] => 2 [2,1] => 1 [1,1,1] => 0

# edit this # code from https://codereview.stackexchange.com/questions/164326 def zigzags(input): input = iter(input) stack = None try: stack = [next(input)] while True: if len(stack) < 2: stack.append(next(input)) else: stack = stack[-2:] a, b = stack if a == b: yield (a,) stack = [b] continue zig = a > b while True: prev = stack[-1] this = next(input) if prev == this or zig == (prev > this): break stack.append(this) zig = not zig yield tuple(stack) stack.append(this) except StopIteration: pass if stack: yield tuple(stack) def statistic(input): return max(len(z) for z in zigzags(input)) # DON'T edit this parent_initializer = lambda x: Words([0,1], x) element_repr = lambda X: str(X) levels = [1, 2, 3, 4, 5, 6] for level in levels: for elt in parent_initializer(level): print('%s => %s' % (element_repr(elt), statistic(elt)))

object_name => value E.g. 0 => 1 1 => 1 00 => 1 01 => 2 10 => 2

# edit this def statistic(P): P = [ list(row) for row in P ] m = min(min(row) for row in P) return sum(row.count(m) for row in P) # DON'T edit this def parent_initializer(n): class MyPlanePartitions(): def __iter__(self): for P in PlanePartitions(n): yield PlanePartition(P[:-1]) def cardinality(self): card = [1, 1, 3, 6, 13, 24, 48, 86, 160, 282, 500, 859, 1479, 2485, 4167, 6879] if n < len(card): return card[n] else: return Infinity return MyPlanePartitions() def PlanePartitions(n, outer=None): emp = [[]] if n == 0: yield [emp] return if outer == emp: yield [] return if outer is None: outer = [n]*n for k in range(1,n+1): for L in Partitions(k, outer=outer): for Pp in PlanePartitions(n-k, outer=L): Pp = [L] + Pp yield Pp element_repr = lambda X: str([Partition(la) for la in X]).replace(" ","") levels = [1, 2, 3, 4, 5, 6, 7] for level in levels: for elt in parent_initializer(level): print('%s => %s' % (element_repr(elt), statistic(elt)))

object_name => value E.g. [[1]] => 1 [[1],[1]] => 2 [[2]] => 1 [[1,1]] => 2 [[1],[1],[1]] => 3

# edit this # please simplify def cyclic_interval(a, b, n): if a <= b: return list(range(a, b+1)) return list(range(a, n+1)) + list(range(1, b+1)) def alignments(x): pi = x.to_signed_permutation().permutation() n = len(pi) al = [] for i, e in enumerate(pi, 1): for j, f in enumerate(pi, 1): if i == j: continue a, b = sorted([i, e]) c, d = sorted([j, f]) if a <= c <= b <= d or c <= a <= d <= b: continue if (e not in cyclic_interval(i, f-1, n) or f not in cyclic_interval(e+1, j, n)): continue if j == f and x[j-1] < 0: continue if i == e and x[i-1] > 0: continue al.append((i, e, j, f)) return al def statistic(x): return len(alignments(x)) # DON'T edit this def parent_initializer(n): class MyDecoratedPermutations(): def __iter__(self): for tau in DecoratedPermutations(n): yield tau def cardinality(self): return sum( factorial(n)/factorial(k) for k in range(n+1) ) return MyDecoratedPermutations() def DecoratedPermutations(n): for sigma in Permutations(n): F = sigma.fixed_points() for X in Subsets(F): tau = list(sigma) for i in X: tau[i-1] = -tau[i-1] yield SignedPermutations(n)(tau) def element_repr(pi): pi = list(pi) for i,a in enumerate(pi): if a == i+1: pi[i] = 0 elif -a == i+1: pi[i] = -1 return str(pi).replace(" ","").replace("0","+").replace("-1","-") levels = [2, 3, 4, 5] for level in levels: for elt in parent_initializer(level): print('%s => %s' % (element_repr(elt), statistic(elt)))

object_name => value E.g. [+] => 0 [-] => 0 [+,+] => 0 [-,+] => 1 [+,-] => 1

# edit this def atomic_length(pi): """ EXAMPLES:: sage: l = [atomic_length(SignedPermutations(n).long_element()) for n in range(1,8)] sage: l sage: fricas.guess(l)[0].sage().factor() 1/6*(4*n + 3)*(n + 2)*(n + 1) """ W = WeylGroup(pi.parent().coxeter_type()) w = W.from_reduced_word(pi.reduced_word()) return sum(a.height() for a in w.inversions(inversion_type="roots")) # DON'T edit this def parent_initializer(n): class MySignedPermutations(): def __iter__(self): for tau in SignedPermutations(n): yield tau def cardinality(self): return 2**n*factorial(n) return MySignedPermutations() element_repr = repr levels = [2, 3, 4] for level in levels: for elt in parent_initializer(level): print('%s => %s' % (element_repr(elt), statistic(elt)))

object_name => value E.g. [1] => 0 [-1] => 1 [1,2] => 0 [1,-2] => 1 [-1,2] => 6

# edit this def fdeg(p): R. = QQ[] f = prod([ 1-q*q^j for j in range(p.size()) ]) b = sum([ j*k for j, k in enumerate(p) ]) return (q^b)*(f/p.hook_polynomial(q,q)) def fake_degree(a): R. = QQ[] if a == 0: return R(0) else: z = SymmetricFunctions(QQ).schur()(a) return R(sum([z.coefficient(p)*fdeg(p) for p in z.support()])) def standard_CSP(p, r): P = parent(p) q = P.gen() PP. = PolynomialRing(QQ) R. = PP.quo(q^r-1) return R(p).lift() def orbit_lengths(p, r): p = standard_CSP(p, r) p = {k: p[k] for k in range(r)} divs = sorted(divisors(r)) O = dict() for d in divs: count = p[r-d] if count: O[r//d] = count if count: for k in range(0, r, d): p[k] -= count assert all(c == 0 for c in p.values()), "%s" % p return O def has_cyclic_permutation_representation(p, r): if p == 0: return True try: lo = orbit_lengths(p, r) except AssertionError: return False return all(v in ZZ and ZZ(v) >= 0 for v in lo.values()) def statistic(la): s = SymmetricFunctions(QQ).schur() f = fake_degree(s(la)) for r in reversed(divisors(la.size())): if has_cyclic_permutation_representation(f, r): return r # alternative implementation using Theorem 2.7 def has_cyclic_permutation_representation_slow(p, r): for k in divisors(r): if sum(moebius(k//j) * p.subs(q=QQbar.zeta(r)^j) for j in divisors(k)) < 0: return False return True def statistic_slow(la): s = SymmetricFunctions(QQ).schur() f = fake_degree(s(la)) for r in reversed(divisors(la.size())): if has_cyclic_permutation_representation_slow(f, r): return r # DON'T edit this parent_initializer = SkewPartitions element_repr = lambda P: str(list(P)).replace(" ","") levels = [2, 3, 4, 5] for level in levels: for elt in parent_initializer(level): print('%s => %s' % (element_repr(elt), statistic(elt)))

object_name => value E.g. [[1],[]] => 1 [[2],[]] => 2 [[1,1],[]] => 1 [[2,1],[1]] => 2 [[3],[]] => 3

# edit this @cached_function def aux(L): if L.cardinality() == 1: return 1, 1 b = L.bottom() U = L.upper_covers(b) J = set() for k in range(1, len(U)+1): for S in Subsets(U, k): J.add(L.join(S)) t = L.top() C = {} for j in J: M = L.sublattice(L.interval(j, t)) M = M.canonical_label() l, c = aux(M) C[l+1] = C.get(l+1, 0) + c l = min(C.keys()) return l, C[l] def statistic(L): return aux(L.canonical_label())[1] # DON'T edit this def parent_initializer(n): class MyLattices(): def __iter__(self): if n <= 2: for P in Posets(n): if P.is_lattice(): yield LatticePoset(P) else: for P in Posets(n-2): Q = P.with_bounds() if Q.is_lattice(): yield LatticePoset(Q) def cardinality(self): l = [1, 1, 1, 1, 2, 5, 15, 53, 222, 1078, 5994, 37622, 262776, 2018305, 16873364, 152233518, 1471613387, 15150569446, 165269824761, 1901910625578, 23003059864006] if n < len(l): return l[n] else: return Infinity return MyLattices() element_repr = lambda X: str( ( sorted(X._hasse_diagram.canonical_label().cover_relations()), len(X._hasse_diagram.vertices(sort=False)) ) ) levels = [3, 4, 5, 6, 7] for level in levels: for elt in parent_initializer(level): print('%s => %s' % (element_repr(elt), statistic(elt)))

object_name => value E.g. ([],1) => 1 ([(0,1)],2) => 1 ([(0,2),(2,1)],3) => 1 ([(0,1),(0,2),(1,3),(2,3)],4) => 1 ([(0,3),(2,1),(3,2)],4) => 1

# edit this from sage.combinat.permutation import to_standard def occurrences(p, pattern): """ Return elements in sequences of distinct blocks order isomorphic to `pattern`. """ o = [] for i, b in enumerate(p): q = p[i+1:] for e in b: o.extend(occurrences_aux(q, [e], pattern)) return o def occurrences_aux(p, prefix, pattern): """ Return all occurrences of the form `(prefix, suffix)` order isomorphic to `pattern`, where `suffix` are elements in sequences of distinct blocks of `p`. """ # TODO: we could do better, probably if len(pattern) == len(prefix): yield prefix else: pattern_prefix = to_standard(pattern[:len(prefix)+1]) for i, b in enumerate(p): q = p[i+1:] for e in b: new_prefix = prefix + [e] if to_standard(new_prefix) == pattern_prefix: yield from occurrences_aux(q, new_prefix, pattern) def statistic(p): return len(occurrences(p, [1,3,2])) # DON'T edit this parent_initializer = OrderedSetPartitions element_repr = repr levels = [2, 3, 4] for level in levels: for elt in parent_initializer(level): print('%s => %s' % (element_repr(elt), statistic(elt)))

object_name => value E.g. [{1}] => 0 [{1},{2}] => 0 [{2},{1}] => 0 [{1,2}] => 0 [{1},{2},{3}] => 0