
Learning proofs for the classification of nilpotent semigroups
Machine learning is applied to find proofs, with smaller or smallest num...
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Typal Heterogeneous Equality Types
The usual homogeneous form of equality type in MartinLöf Type Theory co...
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Bijective recurrences concerning two Schröder triangles
Let r(n,k) (resp. s(n,k)) be the number of Schröder paths (resp. little ...
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An InstanceBased Algorithm for Deciding the Bias of a Coin
Let q ∈ (0,1) and δ∈ (0,1) be real numbers, and let C be a coin that com...
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Extended Irreducible Binary Sextic Goppa codes
Let n (>3) be a prime number and F_2^n a finite field of 2^n elements. L...
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Simple proofs of estimations of Ramsey numbers and of discrepancy
In this expository note we present simple proofs of the lower bound of R...
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Sumsets of Wythoff Sequences, Fibonacci Representation, and Beyond
Let α = (1+√(5))/2 and define the lower and upper Wythoff sequences by a...
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Bijective proofs for Eulerian numbers in types B and D
Let ⟨n k⟩, ⟨B_n k⟩, and ⟨D_n k⟩ be the Eulerian numbers in the types A, B, and D, respectively – that is, the number of permutations of n elements with k descents, the number of signed permutations (of n elements) with k type B descents, the number of even signed permutations (of n elements) with k type D descents. Let S_n(t) = ∑_k = 0^n1⟨n k⟩ t^k, B_n(t) = ∑_k = 0^n ⟨B_n k⟩ t^k, and D_n(t) = ∑_k = 0^n ⟨D_n k⟩ t^k. We give bijective proofs of the identity B_n(t^2) = (1 + t)^n+1S_n(t)  2nt_n(t^2) and of Stembridge's identity D_n(t) = B_n (t)  n2^(n1)tS_n1(t). These bijective proofs rely on a representation of signed permutations as paths. Using this representation we also establish a bijective correspondence between even signed permutations and pairs (w, E) with ([n], E) a threshold graph and w a degree ordering of ([n], E).
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