bibkey: wiseman2018a026010 authors: Gus Wiseman year: 2018 title: “OEIS A026010, a(n) = number of (s(0), s(1), …, s(n)) such that s(i) is a nonnegative integer and |s(i) - s(i-1)| = 1 for i = 1,2,…,n and s(0) = 2” doi: null url: https://oeis.org/A026010 claim: “Conjecture: a(n) is the number of integer compositions of n + 2 in which the even parts appear as often at even positions as at odd positions (confirmed up to n = 19). - Gus Wiseman, Mar 17 2018” strata_touched:

• D5/S3/Combinatorics/BalancedCompositionPaths license: citation-only triage: anchor

OEIS A026010

The entry counts the height sequences of length n + 1 that start at two, move by one at every step and never go below zero. Its terms begin 1, 2, 4, 7, 14, 25, 50, 91, 182, 336, 672, ….

Verified locator

• URL: https://oeis.org/A026010
• Locator: COMMENTS, “Conjecture: a(n) is the number of integer compositions of n + 2 in which the even parts appear as often at even positions as at odd positions (confirmed up to n = 19). - Gus Wiseman, Mar 17 2018”
• Revision read: #55, Oct 13 2025. The comment stands and carries no answer.

Reading of the statement

Positions of a composition are counted from one, so the first part sits at an odd position. A composition with no even part at all is balanced, both counts being zero. The entry’s own worked list fixes the reading: for n = 3 the seven compositions of five are (5), (3,1,1), (1,3,1), (1,1,3), (2,2,1), (1,2,2), (1,1,1,1,1). Note that (2,1,2) is excluded, its two even parts sitting at positions one and three, while (1,2,2) is included.

Scope of the recorded answer

The comment holds for every n. Both sides are windows of the same kernel.

Write w for the unrestricted walk kernel on the integers: w 0 is the indicator of the origin and w (n+1) z = w n (z-1) + w n (z+1). It is even in z, and it is Pascal’s array read in displacement coordinates.

The walk side is a reflection. Sequences from height two that stay nonnegative are all sequences minus those that touch -1, and reflecting across that line matches the offending ones with all sequences from -4. Summing over the end height telescopes, because the subtracted index is exactly three larger than the added one, and three consecutive kernel entries survive.

The composition side is a window of width six. Recursion on the first part splits a composition three ways: a first part at least three loses two and keeps every position, a first part one is deleted and reverses position parity, and a first part two contributes one before being deleted. Tracking the signed difference between even parts at odd and at even positions, the count at difference b is the sum of the six kernel entries centred at 3b, namely from 3b-3 to 3b+2.

At b = 0 that window is w 0 + 2 w 1 + 2 w 2 + w 3 once evenness is used, and the reflected walk sum at height two is the same expression term by term. No generating function, no square root and no real analysis enter.

The bridges to the literal objects are part of the formal statement rather than left implicit: the walk count is the cardinality of an explicit finite set of height sequences, and the composition count is the cardinality of a filter on the compositions of n + 2.

Bounded prior-resolution evidence

The entry was read in full at revision #55 and records no proof and no reference to one. Its cross-references A026009, A050168, A037952, A051924 and A097613 carry no answer either. The partition analogues A300787 and A300788 are different statements and also unproved. Searches for a published proof connecting these compositions to lattice paths returned nothing. Citation indices were not exhaustively reachable, so this is a bounded negative finding and no worldwide priority claim is made.