Ferrers Boards with Prescribed Ordered Level Numbers

Kenneth Barrese, Weighted File Placements on Singleton Boards, arXiv:2312.10855v1, Section 2.2, asks:

This suggests the open question, given an ordered set of level numbers: (l_1, l_2, . . . , l_n), how many unique Ferrers boards have those specific level numbers.

Section 5 repeats this question. For every positive m and n, count every weakly increasing n-tuple of column heights between zero and mn in the fixed n-column, mn-row rectangle, with a prescribed top-down vector l of n nonnegative level totals. A level has m rows; its total counts actual occupied cells. Leading zero columns and empty top levels are retained. The input board need not be singleton, and no permutation of l is allowed.

The same prescribed-level counting problem has an earlier all-width formulation in Kenneth Barrese, Nicholas Loehr, Jeffrey Remmel and Bruce E. Sagan, Bijections on m-level Rook Placements, arXiv:1508.05944v1, Section 6.2:

If C is a singleton board, then perhaps counting the number of Ferrers boards B with l(B) = C would be a good start to this problem, but this too remains open.

For every positive m and every nonempty positive singleton target C, count all normalized Ferrers boards B of arbitrary width with l(B)=C. Definition 4 reverses the nonempty bottom-up level totals. Here normalization uses positive weakly increasing column-height lists: Section 4.1 aligns lower-right cells and treats leading zero padding as another notation for the same board. Arbitrary horizontal translations as distinct unbounded absolute-grid subsets are a different carrier. The bibliographic header identifies the later 2023 question; the original 2015 formulation and its governing conventions are identified explicitly here.

Pre-registration #23382 fixes the complete prescribed-level problem in both source conventions. The joint conclusion uses BarreseOrderedLevels.source2023_count for the fixed rectangle and BarreseSourceNormalization.lOperator_count for the actual native all-width operator fiber. These are two scopes of one registered counting answer, not separate problem credits. The larger question of counting an entire m-level rook equivalence class is outside this answer.

The bounded literature and all-state registration qualification recorded in #23382 found no complete earlier answer or other-owner equivalent registration within its searched scope. The existing singleton representative and rook-equivalence results do not enumerate these fibers. The square-family supplier in PR #14863 concerns a proper subfamily; the rectangular Gaussian polynomial is an ingredient, not a complete prescribed-level solution. These findings do not establish worldwide absence, originality or priority.

Put r=m-1 and write a for bottom-up totals. Let G(N,k) be the finite Gaussian polynomial, defined by G(N,0)=1, G(0,k+1)=0 and G(N+1,k+1)=G(N,k+1)+q^(N-k)G(N,k). Set T_a(u,v)=0 unless v<=u and mv<=a; otherwise set T_a(u,v)=[q^(a-mv)]G(u-v+r,r). For states u,v in {0,...,U}, evaluate F([],u)=1 and F(a::as,u)=sum_(v=0)^U T_a(u,v)F(as,v).

Padded row-column conjugation with horizontal reflection preserves each literal level total. Subtracting a level's final row v from its first m-1 rows gives a rectangle of weight a-mv. The proof directly uses D5/S3/Combinatorics/CylindricPartition/LiUncuRectangle.gauss_rectangle. Compatible level gluing then identifies the finite scalar recurrence with the native board count. The fixed-ambient answer is F(reverse(l),n).

For the all-width problem, positive source columns are bijective to finite Young diagrams. The bottom level total derives width<=last(c), and vanishing above length(c) derives height<=m*length(c). Unique padding therefore gives F(reverse(c),last(c)) without a width premise. The actual trimmed operator equality fiber is bijective to this level-total fiber. If any entry of c is zero, the extended formula is zero. The separate extension l(empty)=[] has one preimage, with U=0; Definition 4 itself requires a largest nonempty level.

A mismatch between a native board fiber and its recurrence value for any positive m and any original ordered data would refute the claimed correspondence. This includes m=1, empty or infeasible data, the all-zero fixed-ambient vector, leading zero slots, top padding, and the all-width bound derived from data. Changing to unordered levels, singleton inputs, a prescribed input width or unbounded absolute translations changes the question.

The seven D5 sources establish the row coordinates, rectangle correspondence, Gaussian transfer, compatible gluing, native fixed-ambient count, derived all-width padding and exact native l-operator fiber count. gauss_rectangle and Mathlib's YoungDiagram.rowLen_ofRowLens are reused directly. The scalar recurrence is over U+1 row-bound states, not over original boards.

All thirteen authored public theorem registrations compile. Four have validated source-binding certificates: BarreseAllWidths.all_width_count, BarreseAllWidths.width_bound, FerrersLevelCounting.paths_card and FerrersLevelGluing.flat_card. Nine audits remain unfinished under #23540: BarreseOrderedLevels.native_count (source.params); BarreseOrderedLevels.source2023_count, BarreseSourceNormalization.lOperator_count and BarreseSourceNormalization.source2015_count (source.dictionary_or_proof_coordinate); FerrersLevelTransfer.level_card (source.unsafe_or_external, site Nat.decLe); FerrersRowCoordinates.board_row_count, conjugate_involutive and diagram_rowLen (source.telescope_reconstruction); and FerrersRowCoordinates.diagram_mem (source.state_binder). These are unfinished source-binding audits, not validated certificates or undecidability results.

theorem; this is the externally printed prescribed ordered level-number counting question. The two terminal declarations cover the complete fixed-ambient and normalized all-width source scopes jointly. Auxiliary modules and consumed coordinate helpers are not additional problems.

Literature and ownership absence are bounded by the completed qualification in #23382. The finite 2015 answer uses the paper's normalization, lower-right alignment and left-zero-padding working convention; it does not count arbitrary absolute horizontal translations separately. The empty native operator is an explicitly declared extension. Nine source-binding audits remain unfinished as listed above. Compilation and mathematical correspondence do not establish worldwide novelty, required CI, merge, first dev push or solved-problem credit.