special1digitinorder

Special1digitinorder is a term encountered mainly in recreational mathematics and puzzle communities to describe a constraint on decimal digit strings related to monotone ordering. The phrase is not tied to a single formal standard, but is used to denote a property involving how digits appear in order.

One common interpretation defines a decimal string as special1digitinorder if exactly one pair of adjacent digits forms a descent, meaning there is exactly one index i such that the i-th digit is greater than the (i+1)-th digit (d[i] > d[i+1]), while all other adjacent pairs satisfy d[j] <= d[j+1]. In this sense, the string is almost nondecreasing, with a single break in the order. For example, the string 1323 has a single descent (3 > 2) and is thus special1digitinorder, whereas 1234 has no descents and would not meet this criterion, and 1432 has two descents (4 > 3 and 3 > 2) and would not.

Variants exist where the property is viewed through different lenses, such as requiring that removing exactly one digit yields a nondecreasing sequence, or counting the minimum number of deletions needed to achieve monotone order and testing whether that minimum equals one. These interpretations influence how the concept is applied in puzzles and algorithms.

Applications of the idea include puzzle design, string-processing exercises, and coding interview problems that test understanding of order-related constraints in digit strings. See also monotone sequences, descents in sequences, and almost-sorted problems.