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Stanley EC1 §1.6.2. For every \(\sigma \in \mathfrak {S}_{n+2}\),

The longest alternating subsequence has length exactly two more than the number of turning points: one element from each “monotone run” of \(\sigma \), plus one boundary element.

Formalization remark. This took five sequential interactive subagent sessions. The naive “leftmost turn in \([\sigma _i, \sigma _{i+2})\)” injection (mapping each subseq sign-flip to a turn) is not injective — adjacent flips can share a turn-window. Session B-4 pivoted from per-interval to a global bound via the subseq-monotonicity of \(\mathrm{flip\_ count}\).

Stanley’s statement (EC1 p. 61):

Let \(S \subseteq [n-1]\). Then \(\beta _n(S) \leq E_n\), with equality if and only if \(S = \{ 1, 3, 5, \dots \} \cap [n-1]\) or \(S = \{ 2, 4, 6, \dots \} \cap [n-1]\).

Here \(E_n\) is the \(n\)-th Euler number, realized as \(\beta (\mathrm{Alt}_n)\).

Our formalization, in contrapositive form: for \(n \geq 2\) and \(D \subseteq \{ 0, \dots , n-1\} \),

The “\(S = \{ 2, 4, 6, \dots \} \)” alternative in Stanley’s statement reduces to the complementation symmetry already proved at the \(\beta \)-level (, see also 1.15): the two alternating descent sets have equal \(\beta \).

\()\mathrm{Alt}_n = \{ \, i \in \{ 0, \dots , n-1\} : i \text{ is odd}\, \} \). Stanley’s “alternating set” is \(S = \{ 1, 3, 5, \dots \} \cap [n-1]\) (eq. 1.65, p. 60); via our index shift this matches \(\mathrm{Alt}_n\) exactly.

For a sequence \(\mathrm{ss}\) of internal positions, \(\texttt{apply\_ psis}(\mathrm{ss})(w)\) is the left-fold composition of the corresponding \(\psi _{i}\). By Theorem 2.11 and Theorem 2.10, the result depends only on the set \(\mathrm{ss} \subseteq \texttt{internal\_ vertices}(w)\), and the M-equivalence class \([w] = \{ \texttt{apply\_ psis}(\mathrm{ss})(w) : \mathrm{ss} \subseteq \texttt{internal\_ vertices}(w)\} \) has size \(2^{\iota (w)}\).

\(\mathrm{as}(\sigma )\) is the length of the longest alternating subsequence of \(\sigma \), formalized as

For \(D \subseteq \{ 0, \dots , n-1\} \),

This is Stanley’s \(\beta _n(S)\) from EC1 eq. (1.32), p. 30.

\(\texttt{cde} \; ::=\; \mathtt{C\_ letter} \; \mid \; \mathtt{D\_ letter} \; \mid \; \mathtt{E\_ letter}\). These correspond to Stanley’s cd-letters \(\mathtt{c}\), \(\mathtt{d}\) and the deleted endpoint \(\mathtt{e}\).

\(u_{w} \; =\; [\, \texttt{is\_ descent\_ seq}(w, k) : k \in \{ 0, \dots , |w| - 2\} \, ]\) is the bit-vector of descent positions.

For a permutation \(\sigma \), this matches Stanley’s \(u_{S} = e_1 e_2 \cdots e_{n-1}\) (eq. 1.60, p. 58) with \(e_i = a\) if \(i \notin S\) and \(e_i = b\) if \(i \in S\). Our representation uses \(\mathtt{false} \mapsto a\) and \(\mathtt{true} \mapsto b\).

Position \(i\) in \(w\) is classified as:

For a permutation \(\sigma \in \mathfrak {S}_{n}\), the cycle count \(c(\sigma )\) is the number of cycles in the disjoint-cycle decomposition of \(\sigma \). In MathComp this is the cardinality of \(\mathrm{porbits}(\sigma )\), the set of orbits of \(\sigma \) acting on \(\{ 0, \dots , n-1\} \) by iteration.

Set \(\operatorname {des}(\sigma ) = |D(\sigma )|\) and \(\operatorname {asc}(\sigma ) = n - \operatorname {des}(\sigma )\).

Let \(\sigma \in \mathfrak {S}_{n+1}\). Position \(i \in \{ 0, \dots , n-1\} \) is a descent of \(\sigma \) when \(\sigma (i+1) {\lt} \sigma (i)\). The descent set is

Let \(\mathrm{alt\_ desc\_ set}\, n \subseteq \{ 0, \dots , n-1\} \) denote the alternating descent pattern \(\{ 0, 2, 4, \dots \} \). Define

\(A_n\) is the number of alternating permutations of \(\mathfrak {S}_{n}\).

For \(n, k \in \mathbb {N}\), the Eulerian number is the count

We write eulerian n k in the formalization, indexed so that \(n\) is “permutation size minus one” and \(k\) ranges over \(0, \dots , n\).

\(\texttt{expand\_ cde}\, m \in \mathrm{seq}\, (\mathrm{seq}\, \mathrm{bool})\) substitutes \(\mathtt{c} \mapsto a + b\), \(\mathtt{d} \mapsto ab + ba\), \(\mathtt{e} \mapsto 1\) in the cd-monomial \(m\):

Each element of the result is a bit-string representing a descent pattern; the total length is \(2^{c} \cdot 2^{d}\) where \(c, d\) are the C- and D-counts of \(m\).

Foata’s first fundamental bijection \(\varphi : \mathfrak {S}_{n+1} \to \mathfrak {S}_{n+1}\) is defined recursively on the seq representation: read \(w = w_1 w_2 \dots w_{n+1}\) left-to-right and at each step splice the new letter into a current word via a cyclic-rotation rule on maximal blocks. Implemented as seq_to_perm \(\circ \) foata \(\circ \) perm_to_seq.

\(\operatorname {hlc}_i(w)\) is true iff in \(M(w)\) the \(\mathrm{Node}\) at position \(i\) has a non-\(\mathrm{Leaf}\) left subtree. This is Stanley’s \(\operatorname {hlc}\) predicate from p. 57 (it distinguishes the cd-letters c and d; see Chapter 3).

\(\texttt{internal\_ vertices}(w)\) is the strictly increasing sequence of positions \(i\) for which \(\operatorname {int}_i(w)\) holds.

\(\mathrm{inv\_ set}(\sigma )\) is the set of inversion pairs; \(\mathrm{inv}(\sigma ) = |\mathrm{inv\_ set}(\sigma )|\) is their count. Stanley writes \(\mathrm{inv}(w)\).

For a sequence \(w\) and \(k \in \mathbb {N}\), \(\texttt{is\_ descent\_ seq}(w, k) = (w_{k+1} {\lt} w_k)\) — the sequence-level analogue of the permutation-level 1.1.

Position \(i\) is internal in \(w\) iff \(i {\lt} |w|\) and \(\operatorname {win}_i(w) {\gt} 1\). Equivalently, the \(\mathrm{Node}\) for position \(i\) in \(M(w)\) has at least one non-\(\mathrm{Leaf}\) child.

Position pair \((i, j)\) is an inversion of \(\sigma \) when \(i {\lt} j\) but \(\sigma (j) {\lt} \sigma (i)\) — i.e., the pair appears in the “wrong” order.

The major index is the sum of (1-indexed) descent positions:

For a non-empty sequence \(s\), \(\texttt{mm\_ pos}(s)\) is the position of \(s\)’s minimum if it occurs strictly before its maximum, and the position of its maximum otherwise. This is the position of the root of \(M(s)\).

A min-max tree over a type \(T\) is the inductive

Its in-order traversal is

\(\texttt{omega\_ seq}(s)\) is the seq-level analogue of \(\omega (D)\): it lists positions \(k\) such that exactly one of \(\{ k, k+1\} \) appears in \(s\). It is the computational backbone of the omega map and the bridge between \(\omega (\) and bit-strings.

For \(D \subseteq \{ 0, \dots , n\} \), the omega set is

This is exactly Stanley’s \(\omega (S)\) from EC1 just before eq. (1.64), p. 60: positions \(k\) where exactly one of \(\{ k, k+1\} \) lies in \(D\). Equivalently, \(\omega (D) = D \, \triangle \, (D - 1)\) where \(D - 1\) is \(D\) shifted by one.

\(\Phi '_{w} = [\, \mathrm{classify\_ vertex\_ cde}(i, w) : i \in \{ 0, \dots , |w|-1\} \, ]\) is the unfiltered list of cd-letters by position. \(\Phi _{w}\) is \(\Phi '_{w}\) with \(\mathtt{E\_ letter}\)s filtered out — Stanley’s \(\Phi _w\), the cd-monomial associated with \(w\) (EC1 pp. 58–59).

For an internal position \(i\) of \(w\), \(\psi _{i}(w)\) is defined by a rank-shift inside the window at \(i\) (the precise formula is given in psi_core.v:219). For non-internal \(i\), \(\psi _{i}\) acts as the identity.

Stanley EC1 §1.4. The bivariate generating function for the joint distribution of \((\mathrm{maj}, \operatorname {des})\) over \(\mathfrak {S}_{n+1}\):

Formalized in \(\mathbb {Z}[q][t]\) (mathcomp’s {poly {poly int}}).

In the polynomial ring \(\mathbb {Z}[q]\):

\(D \subseteq \{ 0, \dots , n-1\} \) is set-alternating iff for every consecutive pair \(i, i+1\) in \(\{ 0, \dots , n-1\} \), exactly one of \(\{ i, i+1\} \) belongs to \(D\). There are exactly two such sets on \(\{ 0, \dots , n-1\} \) (the “odd-position” and “even-position” alternating descents).

For \(D \subseteq \{ 0, \dots , n-1\} \), \(\texttt{set\_ to\_ seq}(D) = \mathrm{sort}\, (\leq , [\, \mathrm{val}(i) : i \in \mathrm{enum}(D)\, ])\) is the sorted list of underlying naturals of elements of \(D\).

\(c(n, k) = \# \{ \sigma \in \mathfrak {S}_{n} : c(\sigma ) = k\} \) counts permutations of \(\{ 0, \dots , n-1\} \) with exactly \(k\) cycles in their disjoint-cycle decomposition. Stanley writes \(c(n, k)\) throughout §1.3.2.

For \(D, E \subseteq \{ 0, \dots , n-1\} \), \(D \, \triangle \, E = (D \setminus E) \cup (E \setminus D)\).

\(D\, \triangle \, \{ i\} = D \, \triangle \, \{ i\} \) flips the membership of position \(i\) in \(D\).

Position \(i \in \{ 0, \dots , n-1\} \) is a turning point of \(\sigma \in \mathfrak {S}_{n+2}\) if the direction of \(\sigma \) changes between slot \(i\) and slot \(i+1\):

\(\mathrm{turn\_ count}(\sigma ) = \# \{ i \in \{ 0, \dots , n-1\} : \mathrm{is\_ turn}(\sigma , i)\} \).

\(\operatorname {win}_i(w)\) is defined by fuel-based recursion that descends \(M(w)\) via \(\texttt{mm\_ pos}\): at the root position one returns \(|w| - i\); otherwise one recurses into the left or right subtree according to whether \(i\) lies before or after the root.

For \(n, k \in \mathbb {N}\), \(\texttt{witness\_ perm}(n, k)\) is an explicit sequence whose cd-monomial is \(c^{k-1}\, d\, c^{n-2-k}\) and whose \(S_w\)-set equals \(\{ k\} \).

This is exactly Stanley’s witness from the proof of Proposition 1.6.4 (EC1 p. 61):

“If \(i \in \omega (T) \setminus \omega (S)\) then let \(\Phi _{w} = c^{i-1}\, d\, c^{n-2-i}\), so \(S_w = \{ i\} \).”

\(\beta (\emptyset ) = 1\) (only the identity has empty descent set) and \(\beta (\{ 0, \dots , n-1\} ) = 1\) (only the order-reversing permutation has full descent set).

\(\beta (D) = \beta (\, \rev _n \cdot D^{c}\, )\), where \(D^{c}\) denotes the set complement and \(\rev _n\) acts pointwise. In particular the two alternating descent sets have equal \(\beta \)-value.

\(\# \{ (i, j) \in \{ 0, \dots , m-1\} ^2 : i {\lt} j\} = \binom {m}{2}\).

Each fixed point is a 1-cycle, so \(c(\mathrm{id}) = n\).

Every cycle is non-empty, so the cycle count is bounded by \(n\).

\(\operatorname {des}(\mathrm{id}_{\mathfrak {S}_{n+1}}) = 0\).

Let \(\rev _n \in \mathfrak {S}_{n+1}\) be the order-reversing permutation \(i \mapsto n - i\). Position \(i\) is a descent of \(\sigma \) if and only if \(i\) is not a descent of \(\rev _n \cdot \sigma \), hence \(\operatorname {des}(\rev _n \cdot \sigma ) = n - \operatorname {des}(\sigma )\). In particular \(\operatorname {des}(\rev _n) = n\).

\(A_{n+1,1} = A_{n+1,n+1} = 1\), and for \(0 \leq k \leq n\), \(A_{n+1,k+1} = A_{n+1,(n-k)+1}\).

\(\sum _{k=0}^{n} A_{n+1,k+1} = (n+1)!\).

\(\mathrm{inv}(\mathrm{id}) = 0\).

\(\mathrm{inv}(\sigma ) \leq \binom {n+1}{2}\). The maximum is achieved by the order-reversing permutation, which has every pair inverted.

\(\mathrm{maj}(\mathrm{id}) = 0\).

\(\mathrm{maj}(\sigma ) \leq \binom {n+1}{2}\). The maximum is achieved by the order-reversing permutation, where every position is a descent and the sum is \(1 + 2 + \dots + n = \binom {n+1}{2}\).

\(k \in \omega (D)\) iff toggling \(D\) at one of \(\{ k, k+1\} \) flips the membership at the other position.

For non-empty \(s\), \(\texttt{mm\_ pos}(s) {\lt} |s|\).

For \(D \subseteq \{ 0, \dots , m\} \) with \(D\) not set-alternating, \(\omega (D) \subsetneq \mathrm{set\_ T}\).

This is the contrapositive of the “\(\Rightarrow \)” direction of Stanley’s eq. (1.65): non-alternating sets have strictly smaller omega. Together with 4.12 it gives the full characterization

For \(m \geq 1\), \(\omega ((\mathrm{Alt}_{m+2})) \; =\; \{ 0, \dots , m\} \; =\; \mathrm{set\_ T}.\) This is the “\(\Leftarrow \)” direction of Stanley’s eq. (1.65) (p. 60): the alternating descent set has full omega.

For \(k \in \{ 0, \dots , m-1\} \) and \(D \subseteq \{ 0, \dots , m\} \),

This identifies the finset-level \(\omega (\) with the seq-level \()\texttt{omega\_ seq}\), letting the heavy machinery in \(\texttt{psi\_ cdindex\_ *.v}\) apply to \(\{ \, \texttt{set}\, \} \)-style consumers.

For every \(\mathrm{ss}\), every position \(i\), and every uniq \(w\): \(\operatorname {win}_i(\texttt{apply\_ psis}(\mathrm{ss})(w)) = \operatorname {win}_i(w)\), \(\operatorname {hlc}_i(\texttt{apply\_ psis}(\mathrm{ss})(w)) = \operatorname {hlc}_i(w)\), \(\operatorname {int}_i(\texttt{apply\_ psis}(\mathrm{ss})(w)) = \operatorname {int}_i(w)\), and the lists of internal vertices coincide.

These four invariances express that \(\Psi \)-action permutes labels inside windows but preserves the unlabelled tree shape — the load-bearing geometric fact behind Stanley Fact #1’s well-definedness clause for \(\Phi _{w}\) (Chapter 3).

\(\sum _{k=0}^{n} c(n, k) = n!\). Each permutation is counted by exactly one \(k\), so the row sum recovers the total.

\((D\, \triangle \, \{ i\} )\, \triangle \, \{ i\} = D\).

\(\texttt{witness\_ perm}(n, k)\) is a uniq sequence of length \(n\) whose multiset of values is \(\{ 0, \dots , n-1\} \). Combined with 5.1, this gives a permutation \(\sigma _{n, k} \in \mathfrak {S}_{n}\) with the prescribed cd-monomial.

Stanley’s statement (verbatim, EC1 p. 60):

Let \(S, T \subseteq [n-1]\). If \(\omega (S) \subsetneq \omega (T)\), then \(\beta _n(S) {\lt} \beta _n(T)\).

Our formalization, for \(D, E \subseteq \{ 0, \dots , m\} \):

\((\mathrm{turn\_ count}(\sigma )).+2 \leq \mathrm{as\_ perm\_ max}(\sigma )\).

Witness: \(I = \{ 0, n+1\} \cup \{ (\mathrm{val}\, t).+1 : t \text{ is a turning point of }\sigma \} \). Cardinality is \(\mathrm{turn\_ count}(\sigma ) + 2\) (boundary endpoints plus one position per turn). Alternation is verified between consecutive picks via the inter-turn monotonicity lemma applied to the no-turn slot interval between them.

\(\mathrm{as\_ perm\_ max}(\sigma ) \leq (\mathrm{turn\_ count}(\sigma )).+2\).

Proved via the global bound chain: flip_count (number of direction changes in a boolean sequence) is monotone under subsequence, so \(\mathrm{flip\_ count}(\mathrm{sign\_ seq}(\mathrm{pick\_ seq}\, \sigma \, I)) \leq \mathrm{flip\_ count}(\mathrm{sign\_ seq}(\mathrm{perm\_ seq}\, \sigma ))\). The right-hand side equals \(\mathrm{turn\_ count}(\sigma )\) via the bridge .

For each \(0 \leq k \leq n\),

Aggregating \(\beta \) over descent sets of fixed size recovers the Eulerian number.

For every \(n \in \mathbb {N}\),

Proof status. Derived in one line from two_eulerA_split (which uses beta_compl from beta_swap.v) and the named admit:

Sanity. The base case \(n = 0\) (i.e. \(2 A_2 = 2\)) is proven unconditionally as euler_rec_n0_direct (no admit used). The cases \(n = 1, 2\) are stated via euler_rec and inherit the admit transitively.

In \(\mathbb {Z}\),

Proved by Worpitzky inversion using an alternating-binomial Kronecker identity.

For all \(n, k \in \mathbb {N}\),

Proof is by the value-based “insert max” bijection \(\mathfrak {S}_{n+2} \simeq \mathfrak {S}_{n+1} \times \{ 0, \dots , n+1\} \).

Stanley’s statement (verbatim, p. 58):

Let \(w \in \mathfrak {S}_n\), and let \([w]\) be the M-equivalence class containing \(w\). Then \(\sum _{v \in [w]} u_{D(v)} = \Phi _{w}(a + b,\; ab + ba)\).

Our formalization, with multisets represented as sorted seqs:

Stanley uses Fact #3 as a stepping stone to Theorem 1.6.3 (the ab-index of \(\mathfrak {S}_n\) factors through the cd-substitution); our \(\texttt{fact3}\) matches this directly.

\(\varphi \) is injective (hence bijective on the finite group \(\mathfrak {S}_{n+1}\)). Proof: explicit construction of the inverse \(\varphi ^{-1}\) on seqs (cuts at \(P\)-elements rather than after them, undoing the cyclic rotation block-wise) plus a cancellation lemma \(\varphi ^{-1}(\varphi (w)) = w\).

For every \(\sigma \in \mathfrak {S}_{n+1}\),

This is the load-bearing identity of the bijection. It is proved via the seq-level invariant foata_inv_eq_maj: \(\mathrm{inv\_ seq}(\mathrm{foata}(w)) = \mathrm{maj\_ seq}(w)\), which in turn is proved by reverse induction on \(w\) using the per-step invariant foata_step_inv (a tally of cyclic-rotation effects on the inversion count).

Stanley EC1 Theorem 1.3.4 (MacMahon). For every \(n, k \in \mathbb {N}\),

The proof is a direct consequence of 6.18 and 6.19: \(\varphi \) is a bijection of \(\mathfrak {S}_{n+1}\) that maps \(\{ \sigma : \mathrm{maj}(\sigma ) = k\} \) onto \(\{ \sigma : \mathrm{inv}(\sigma ) = k\} \) for every \(k\). The closed form for both distributions is the \(q\)-factorial (6.22 and 6.23).

Stanley EC1 §1.3.4 / Cor 1.3.10. For every \(n \in \mathbb {N}\),

Proved by induction on \(n\) using the value-based “insert \(\mathrm{ord}_{\max }\)” bijection \(\mathfrak {S}_{n+2} \simeq \mathfrak {S}_{n+1} \times \{ 0, \dots , n+1\} \) from \(\texttt{eulerian.v}\). The key step is \(\mathrm{inv}(\mathrm{insert\_ max\_ perm}(\tau , p)) = \mathrm{inv}(\tau ) + (n+1 - p)\).

Let \(\rev _n \in \mathfrak {S}_{n+1}\) be the order-reversing permutation. Then

Each pair \((i, j)\) with \(i {\lt} j\) is either an inversion or a co-inversion of \(\sigma \) (exactly one), and the bijection \((i, j) \mapsto (\rev _n(j), \rev _n(i))\) swaps the two.

Stanley EC1 §1.3.4. For every \(n \in \mathbb {N}\),

Immediate corollary of 6.22 via Foata’s bijection.

For \(\rev _n \in \mathfrak {S}_{n+1}\),

The naive identity \(\mathrm{maj}(\rev _n \cdot \sigma ) = \binom {n+1}{2} - \mathrm{maj}(\sigma )\) is false in general: the asymmetry comes from \(\mathrm{maj}\)’s 1-indexed positions. The correct identity above accounts for it via the per-descent offset \((n - i) + (i + 1) = n + 1\).

For every \(s \in \mathrm{seq}\, \mathbb {N}\),

This identifies \(M(s)\) as a faithful representation of the underlying sequence: every position in \(w\) corresponds to a \(\mathrm{Node}\) of \(M(w)\).

For every \(\mathrm{ss}\) and every uniq \(w\), \(\Phi _{\texttt{apply\_ psis}(\mathrm{ss})(w)} = \Phi _{w}\).

Equivalently: \(\Phi _{w}\) depends only on the M-equivalence class \([w]\) — equivalently, only on \(M(w)\) as an unlabelled tree. This is the well-definedness clause of Stanley’s Fact #1 (EC1 p. 57). Internally it follows from the shape-invariance lemmas in 2.13.

For every uniq \(w\) with \(|w| \geq 2\) and every bit-string \(X\) with \(|X| = |w| - 1\):

This formalizes Stanley’s identity at the top of EC1 p. 61 inside the proof of Prop 1.6.4.

The supporting bit-level definitions (\(\texttt{cde\_ width}\), \(\texttt{cde\_ total\_ width}\), \(\texttt{cde\_ offset}\), \(\texttt{D\_ offsets}\), \(\texttt{expand\_ cde\_ mem\_ iff}\)) live in .

For all \(i, j \in \mathbb {N}\) and every uniq sequence \(w\), \(\psi _{i}(\psi _{j}(w)) = \psi _{j}(\psi _{i}(w))\). This is the commutativity half of Stanley’s Fact #1 (EC1 p. 57): the \(\psi _{i}\) for internal vertices generate an abelian group \(G_w \cong (\mathbb {Z}/2)^{\iota (w)}\), where \(\iota (w)\) is the number of internal vertices.

For every \(i \in \mathbb {N}\) and every uniq sequence \(w\), \(\psi _{i}(\psi _{i}(w)) = w\).

\(A_n(1, t) = \mathrm{eul\_ pol}_n(t) := \sum _{k=0}^{n} A_{n+1, k+1} \cdot t^k\). Setting \(q = 1\) drops the \(\mathrm{maj}\) weight, leaving the classical Eulerian polynomial \(\sum _\sigma t^{\operatorname {des}(\sigma )}\). Formalized via the rmorphism wrapper \(\mathrm{horner\_ eval}(1)\) applied coefficient-wise across the outer polynomial.

\(A_n(q, 1) = [n+1]_q!\). Setting \(t = 1\) collapses the bivariate to the marginal generating function for \(\mathrm{maj}\), which is the q-factorial by 6.23.

For all \(n, k \in \mathbb {N}\),

Combinatorially: \(n+1\) either becomes a fixed point of a \(k\)-cycle permutation (giving \(c(n, k)\)), or is inserted into one of \(n\) positions of a \((k+1)\)-cycle permutation (giving \(n \cdot c(n, k+1)\)).

Formalization route. The proof factors through a construction \(\mathrm{insert\_ after}\, j_0\, s : \mathfrak {S}_{n+1}\) that splices \(\mathrm{ord\_ max}\) into the \(s\)-cycle through \(j_0\). The key algebraic identity

reduces cycle-count preservation to MathComp’s porbits_mul_tperm composed with the fixed-point cycle-count lemma in .

\(\sum _{D \subseteq \{ 0, \dots , n-1\} } \beta (D) = (n+1)!\).

For all \(n, x \in \mathbb {N}\),