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 Incoming links: A Survey of Alternating Permutations by Richard P. Stanley

Entringer number Created 2026-10-20 Updated 2026-11-05

With the convention used here,

E (n, k)

counts the alternating permutations

a_{1} \dots a_{n + 1}

of

{1, \dots, n + 1}

satisfying

a_{1} > a_{2} < a_{3} > a_{4} < \dots, a_{1} = k + 1, 0 \leq k \leq n .

The direction of the first comparison and the shift in the first-entry index are essential when comparing tables. The recurrence relation is

E (0, 0) = 1, E (n, 0) = 0 (n > 0), E (n, k) = E (n, k - 1) + E (n - 1, n - k) (1 \leq k \leq n) .

Hence the first rows in increasing

k

are

(1)

,

(0, 1)

,

(0, 1, 1)

and

(0, 1, 2, 2)

. For

n = 2

, the two alternating permutations are

213

and

312

, explaining the two nonzero entries. The row sum counts all alternating permutations on

n + 1

labels; the last entry counts those on

n

labels. Reversed-row prefix sums compute the Entringer triangle without enumerating the permutations individually.^[ref]

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