bibkey: bala2022a004123 authors: Peter Bala year: 2022 title: “OEIS A004123, number of generalized weak orders on n points” doi: null url: https://oeis.org/A004123 claim: “A004123 NAME: Number of generalized weak orders on n points. Conjecture: Let k be a positive integer. The sequence obtained by reducing a(n) modulo k is eventually periodic with the period dividing phi(k) = A000010(k). For example, modulo 7 we obtain the sequence [1, 2, 3, 4, 2, 0, 0, 2, 3, 4, 2, 0, 0, 2, 3, 4, 2, 0, 0, …] with an apparent period of 6 = phi(7) starting at a(2). Cf. A000670.” strata_touched:
- D5/S1/Recurrence/Periodic/StirlingTransformTotientPeriod license: citation-only triage: anchor
OEIS A004123
The A004123 comment is attributed to Peter Bala, July 8, 2022, and is the
conjecture quoted above; the sibling entry A064618 carries its own note,
bala2018a064618. Both finite formulas use the Stirling numbers of the second
kind S(n,k):
- A064618, offset zero:
a(n) = Sum_{k=0}^n (k!)^2 * S(n,k). - A004123, offset one:
a(n) = Sum_{k=0}^{n-1} 2^k * k! * S(n-1,k)forn >= 1.
The module defines T(w,n) = Sum_{k=0}^n w(k) * k! * S(n,k) over the integers.
It proves T(w,n+phi(m)) = T(w,n) modulo every positive m when n >= m,
for every integer weight w. Terms with k >= m vanish modulo m;
inclusion-exclusion gives a fixed sum of powers. Euler’s theorem for unit
bases and vanishing powers for non-unit bases give the period at each prime
power dividing m, and prime-power divisibility combines the congruences.
The weights w(k)=k! and w(k)=2^k give the two assertions. In the entries’
own indexing, the proved onsets are respectively n >= m and n >= m+1.
Neither onset nor period is asserted to be minimal. The broader A004123
conjecture about every e.g.f. G(exp(x)-1) is outside these two assertions.
Verified locator
- A064618 NAME, OFFSET, FORMULA, and COMMENT: https://oeis.org/A064618
- A004123 NAME, OFFSET, FORMULA, and COMMENT: https://oeis.org/A004123