Verify Row Sum Identity for Stirling Numbers — verify_stirling_row

Verifies that the row sum of Stirling numbers equals J! for each row. This is the fundamental identity: $\sum_{k=1}^{J} |s(J,k)| = J!$

Usage

verify_stirling_row_sum(
  logS,
  J_values = 2:10,
  tolerance = 1e-10,
  verbose = TRUE
)

Arguments

logS: Pre-computed log-Stirling matrix from compute_log_stirling.
J_values: Vector of J values to verify (default: 2:10).
tolerance: Numerical tolerance for comparison (default: 1e-10).
verbose: Logical; if TRUE, print verification results.

Value

Logical indicating whether all verifications passed.

Details

The unsigned Stirling numbers of the first kind satisfy: $$\sum_{k=1}^{J} |s(J,k)| = J!$$

This identity follows from the fact that $|s(J,k)|$ counts permutations of J elements with exactly k cycles, and the total number of permutations is J!.

Examples

logS <- compute_log_stirling(15)
verify_stirling_row_sum(logS, J_values = 2:10)
#> PASS: sum_k |s(2,k)| = 2! [error = 0.00e+00]
#> PASS: sum_k |s(3,k)| = 3! [error = 0.00e+00]
#> PASS: sum_k |s(4,k)| = 4! [error = 4.44e-16]
#> PASS: sum_k |s(5,k)| = 5! [error = 0.00e+00]
#> PASS: sum_k |s(6,k)| = 6! [error = 8.88e-16]
#> PASS: sum_k |s(7,k)| = 7! [error = 0.00e+00]
#> PASS: sum_k |s(8,k)| = 8! [error = 0.00e+00]
#> PASS: sum_k |s(9,k)| = 9! [error = 0.00e+00]
#> PASS: sum_k |s(10,k)| = 10! [error = 0.00e+00]
#> [1] TRUE