Eulerian number

Descent count anchor

Adopt the ordinary descent interpretation as the draft's working definition and record evidence for each imported interpretation.

1. Does the source count permutations of n distinct ordered elements with exactly k positions i satisfying sigma(i) > sigma(i+1)? definition
2. Which read definition establishes that the supplied notation denotes this count? provenance

Interpretation compatibility

Require an explicit translation or separate family declaration before reusing an imported count.

1. If the source uses ascents, excedances or runs, what argument and index translation connect its count to the adopted convention? boundary
2. Does the source introduce repeated elements, signs or an order parameter that changes the counted family? boundary

Column meaning

Record whether a column labels the statistic directly or uses an offset.

1. What do the source's first and second indices mean, and where does column numbering begin? definition
2. Does translating the source's n=3 row recover canonical columns k=0,1,2 with values 1,4,1? measurement

Boundary contract

Adopt A(0,0)=1; for n>=1, support is 0<=k<=n-1. Extend other integer k values by zero; reject negative or noninteger n. [NIST DLMF](https://dlmf.nist.gov/26.14)

1. Does the implementation preserve the empty-permutation value separately from positive-size rows? boundary
2. Will an out-of-support integer k return zero while malformed or unsupported indices produce an explicit error? action

Recurrence contract

For n>=2, A(n,k)=(k+1)A(n-1,k)+(n-k)A(n-1,k-1). [NIST DLMF recurrence](https://dlmf.nist.gov/26.14.E8)

1. Which base rows and zero-extension rules initialize the recurrence? definition
2. Can the selected arithmetic represent every intermediate count exactly, or must the result be labelled approximate or modular? action

Row invariants

For n>=1, require A(n,k)=A(n,n-1-k) and sum over k=0 through n-1 equal to n!. [NIST DLMF identities](https://dlmf.nist.gov/26.14#iii)

1. Does the complete computed row satisfy symmetry and the factorial total? measurement
2. Which small-case enumeration or independently evaluated formula checks the claimed cell beyond these necessary row tests? provenance

Exponent convention

For this model's interface, define P_n(t)=sum_k A(n,k)t^k and require explicit translation from other normalizations.

1. Does the referenced polynomial attach A(n,k) to t^k, t^(k+1) or another expression? definition
2. What exponent shift or normalization must the agent apply before extracting or exporting coefficients? action

Uniform sampling condition

For a uniformly sampled permutation, the definition implies Pr(k descents)=A(n,k)/n!; record this as a derived probability.

1. Are all n! permutations equally likely, or does the task impose weights or restrictions? boundary
2. Should the derived probability remain an exact rational or be rounded with a stated precision? action

Check these first

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these check these first hold for the sense of Eulerian number this model covers, and on what evidence? provenance

Kinds and varieties

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these kinds and varieties hold for the sense of Eulerian number this model covers, and on what evidence? provenance

Identifiers and schemes

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these identifiers and schemes hold for the sense of Eulerian number this model covers, and on what evidence? provenance

Real-world use

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these real-world use hold for the sense of Eulerian number this model covers, and on what evidence? provenance

Failure modes and hazards

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these failure modes and hazards hold for the sense of Eulerian number this model covers, and on what evidence? provenance

Neighbouring kinds and how to tell them apart

Recalled without web access and unsourced; every item is a lead to verify.

1. Which of these neighbouring kinds and how to tell them apart hold for the sense of Eulerian number this model covers, and on what evidence? provenance