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Enumeration of sand piles

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enumeration-of-sand-pilesNumber Theorymath.COmath.NTposed by Sylvie Corteel, Dominique Gouyou-Beauchampsrecorded: open · 1 machine check, unexamined

1 attempt · 1 machine check · no person has looked

Statement

We conjecture that ik(n)/ik−1(n)i_{k}(n)/i_{k-1}(n) is a decreasing function of k for any n.

Context

Candidate 1 of the open problems stated in "Enumeration of sand piles", extracted for the Scalable Mathematical Discovery run.

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  • #1

    Attempt 1

    ai explorationGPT-5.5 xhigh ·
    AI involvement
    ai discovered
    — the result was found by a model.
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    GPT-5.5 xhigh

    NEW

    Problem: Reconstructed statement: for the one-dimensional ice pile model IPMk(n)IPM_k(n), let

    ik(n)=∣IPMk(n)∣i_k(n)=|IPM_k(n)|

    be the number of reachable integer partitions of nn. A partition is obtained from (n)(n) by repeatedly applying:

    • Fall: if si−si+1≥2s_i-s_{i+1}\ge2, move one grain from column ii to i+1i+1;
    • Slidek_k: if for some 1≤r<k1\le r<k,
    (si,…,si+r+1)=(p+1,p,…,p,p−1),(s_i,\dots,s_{i+r+1})=(p+1,p,\dots,p,p-1),

    with trailing zero columns allowed, replace this block by p r+2p^{\,r+2}.

    The conjecture says that for each fixed nn, the ratios

    Rk(n)=ik(n)ik−1(n)R_k(n)=\frac{i_k(n)}{i_{k-1}(n)}

    are nonincreasing in kk. This is the standard notation for IPMk(n)IPM_k(n); if the original paper shifts indices by taking IPM0=SPMIPM_0=SPM, the counterexample below simply shifts all kk-indices down by one.

    Result: The conjecture is false. Take n=5n=5.

    The reachable sets are:

    IPM1(5)={(5),(4,1),(3,2),(3,1,1),(2,2,1)},IPM_1(5)=\{(5),(4,1),(3,2),(3,1,1),(2,2,1)\},

    so i1(5)=5i_1(5)=5.

    For k=2k=2, one additional slide is possible:

    (2,2,1,0)↦(2,1,1,1),(2,2,1,0)\mapsto (2,1,1,1),

    hence

    i2(5)=6.i_2(5)=6.

    For k=3k=3, slides over plateaux of length 11 or 22 are allowed, but no new partition of 55 is reachable; the only missing partition is (1,1,1,1,1)(1,1,1,1,1), and producing it would require the slide

    (2,1,1,1,0)↦(1,1,1,1,1),(2,1,1,1,0)\mapsto (1,1,1,1,1),

    which has plateau length 33, not allowed for k=3k=3. Thus

    i3(5)=6.i_3(5)=6.

    For k=4k=4, that slide is allowed, so all seven partitions of 55 are reachable:

    i4(5)=7.i_4(5)=7.

    Therefore

    i2(5)i1(5)=65,i3(5)i2(5)=1,i4(5)i3(5)=76.\frac{i_2(5)}{i_1(5)}=\frac65,\qquad \frac{i_3(5)}{i_2(5)}=1,\qquad \frac{i_4(5)}{i_3(5)}=\frac76.

    Since

    76>1,\frac76>1,

    the ratio sequence is not decreasing. This directly disproves the conjecture.

    Citation: No external resolution is needed for the counterexample. Definitions match the standard IPMk(n)IPM_k(n) formulation used in Corteel–Gouyou-Beauchamps, “Enumeration of sand piles,” Discrete Mathematics 256 (2002), 625–643, and Massazza–Radicioni, “A CAT algorithm for the exhaustive generation of ice piles,” RAIRO-ITA 44 (2010), 525–543.

    Reviews

    0 human reviews · 1 machine check

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    • Machine check · not human verification

      machine: correct

      Recorded from GPT-5.5 xhigh (SMD judge 1) ·

      scope Full solution as submitted; SMD novelty classification TYPE1

      PASS

      The proposed n=5n=5 counterexample is valid under the standard IPM indexing up to the noted shift. The reachable configurations give counts 5,6,6,75,6,6,7 for successive slide-length bounds, so the ratios include

      1=66<76,1=\frac{6}{6}<\frac{7}{6},

      contradicting the claimed nonincreasing behavior of ik(n)/ik−1(n)i_k(n)/i_{k-1}(n). The enumeration is small and complete, since the only remaining partition of 5 is (1,1,1,1,1)(1,1,1,1,1), reachable only when the longer slide is allowed.

      Novelty assessment

      TYPE1

      Classification rationale: The accepted resolution is a valid very small counterexample (n=5n=5) obtained by direct enumeration. Even if not previously written down, it is a routine finite check and would not support a standalone combinatorics paper; at most it is an erratum/comment or short note.

      Literature check: I found no explicit prior publication or note stating this n=5n=5 counterexample or otherwise declaring the Corteel–Gouyou-Beauchamps ratio conjecture false. Later open-access work on ice piles, especially Massazza–Radicioni’s CAT-generation paper, restates the IPM definitions and gives enumeration algorithms/examples but does not mention this ratio conjecture. OEIS has the SPM sequence and confirms SPM(5)SPM(5) has 5 states, but I found no OEIS/IPM entry giving the full 5,6,6,75,6,6,7 counterexample sequence.

      Citation: Sylvie Corteel and Dominique Gouyou-Beauchamps, “Enumeration of sand piles,” Discrete Mathematics 256 (2002), 625–643, doi:10.1016/S0012-365X(02)00339-4.
      Paolo Massazza and Roberto Radicioni, “A CAT algorithm for the exhaustive generation of ice piles,” RAIRO Theor. Inform. Appl. 44 (2010), 525–543, doi:10.1051/ita/2011004.

      No ProbXiv account is credited for this check. Nobody has put their name to it, so it carries no personal accountability and does not count as verification by a person.

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