We prove a conjecture of R. Ford stated in 2017 in the OEIS entry A000010 (Euler's totient function): for n > 1, the number of semi-meander solutions for n whose top arches contain exactly 2 mountain ranges and exactly 2 arches of length 1 equals phi(n). In Ford's setting, the points 1, ..., 2n lie on a line, the botto...
Roberto Blanco Gómez· Zenodo (CERN European Organi...· 0 citations
We prove a conjecture recorded by J. M. Campbell in 2018 in the OEIS entry A001405 (the central binomial coefficients binomial(n, floor(n/2))). Let NSym be the algebra of noncommutative symmetric functions of Gelfand, Krob, Lascoux, Leclerc, Retakh and Thibon, and let C_n(R, H) be the transition matrix from the ribbon...
Roberto Blanco Gómez· Zenodo (CERN European Organi...· 0 citations
We prove a statement recorded by H. Gruber in 2012 in the OEIS entry A000079 (the powers of 2), verified by him up to n = 17: for n >= 1, the number of distinct finite languages over a one-letter alphabet whose minimum regular expression has alphabetic width n is 2^n. The alphabetic width of a regular expression is the...
Roberto Blanco Gómez· Zenodo (CERN European Organi...· 0 citations
We prove a statement recorded by H. Gruber in 2012 in the OEIS entry A000079 (the powers of 2), verified by him up to n = 17: for n >= 1, the number of distinct finite languages over a one-letter alphabet whose minimum regular expression has alphabetic width n is 2^n. The alphabetic width of a regular expression is the...
Roberto Blanco Gómez· Zenodo (CERN European Organi...· 0 citations
We prove a conjecture recorded by J. M. Campbell in 2018 in the OEIS entry A001405 (the central binomial coefficients binomial(n, floor(n/2))). Let NSym be the algebra of noncommutative symmetric functions of Gelfand, Krob, Lascoux, Leclerc, Retakh and Thibon, and let C_n(R, H) be the transition matrix from the ribbon...
Roberto Blanco Gómez· Zenodo (CERN European Organi...· 0 citations
We prove a conjecture of T. Piesk stated in 2013 in the OEIS entries A002487 (Stern's diatomic sequence) and A223541 (the array of nim-products 2^m ⊗ 2^n): for every n >= 0, the number of distinct values of 2^x ⊗ 2^y with x + y = n equals A002487(n+1), the number of hyperbinary representations of n. Here ⊗ is Conway's...
Roberto Blanco Gómez· Zenodo (CERN European Organi...· 0 citations
We prove a conjecture of T. Piesk stated in 2013 in the OEIS entries A002487 (Stern's diatomic sequence) and A223541 (the array of nim-products 2^m ⊗ 2^n): for every n >= 0, the number of distinct values of 2^x ⊗ 2^y with x + y = n equals A002487(n+1), the number of hyperbinary representations of n. Here ⊗ is Conway's...
Roberto Blanco Gómez· Zenodo (CERN European Organi...· 0 citations
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