Wang transform inequality

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Minc’s list · Conjecture 4

Wang transform inequality

Proved by us Preprint, September 2026

Posed by
Wang
First posed
1977

Statement

Posed by E. T. H. Wang in 1977, in the paper where he proved the order-three case of Conjecture 3; it is Conjecture 4 in Minc’s catalogue.

Conjecture 4. Let \(\Omega_n\) be the set of \(n\times n\) doubly stochastic matrices and let \(J_n\) be the \(n\times n\) matrix with every entry \(1/n\). If \(A\in\Omega_n\) and \(n\ge 2\), then

\[ \operatorname{per}A\;\ge\;\operatorname{per}\!\left(\frac{nJ_n+A}{n+1}\right). \]

If \(n\ge 3\), equality holds if and only if \(A=J_n\).

The matrix \((nJ_n+A)/(n+1)\) lies on the segment from \(J_n\) to \(A\), at distance \(1/(n+1)\) of the way from \(J_n\). Compare Conjecture 3, which uses \((nJ_n-A)/(n-1)\) instead.

Resolution

We proved the conjecture in every order, with equality only at \(A=J_n\); this includes \(n=2\), which the equality clause above leaves out.

For \(n\ge2\), the proof compares two global estimates in terms of \(r=\lVert A-J_n\rVert_F\). Both rest on our proof of Conjecture 3, which gives the lower bound

\[ \operatorname{per}A\ \ge\ \frac{n!}{n^n}\exp\!\left(\Bigl(\frac1{2n}+\frac1{3n^2}\Bigr)r^2\right) \]

and an estimate for sums of subpermanents of matrices with zero line sums. Applied to the expansion of the permanent around \(J_n\), that estimate gives

\[ \operatorname{per}\!\left(\frac{nJ_n+A}{n+1}\right)\ \le\ \frac{n!}{n^n}\bigl(1+b_nr^2\bigr) \]

with an explicit constant \(0<b_n<1/(2n)\). The two bounds separate whenever \(A\ne J_n\), with the quantitative gap \(\operatorname{per}A-\operatorname{per}\bigl((nJ_n+A)/(n+1)\bigr)\ge\frac{n!}{n^n}\bigl(\frac1{2n}+\frac1{3n^2}-b_n\bigr)r^2\).

Read our paper →

Earlier progress

  • Wang (1977) proved the case \(n=3\); Lih and Wang (1981) gave another proof by showing that the permanent increases along every segment from \(J_3\).
  • Chang (1983) proved the inequality for all \(A\) outside a sufficiently large neighbourhood of \(J_n\).
  • Chang (1988) and Foregger (1988) independently proved the case \(n=4\).
  • Hwang (1986, 1989) proved it for partly decomposable matrices, for symmetric positive semidefinite matrices, and for several structured families, including matrices with \(n-1\) identical rows.
  • The inequality holds whenever the permanent is nondecreasing along the segment from \(J_n\) to \(A\), the property studied in Problem 8.

References

  1. E. T. H. Wang, On a conjecture of M. Marcus and H. Minc, Linear and Multilinear Algebra 5 (1977), 145–148. doi:10.1080/03081087708817189
  2. K.-W. Lih and E. T. H. Wang, Monotonicity conjecture on permanents of doubly stochastic matrices, Proc. Amer. Math. Soc. 82 (1981), 173–178. doi:10.1090/S0002-9939-1981-0609645-5
  3. D. K. Chang, Notes on permanents of doubly stochastic matrices, Linear and Multilinear Algebra 14 (1983), 349–356. doi:10.1080/03081088308817570
  4. S.-G. Hwang, The monotonicity of and the Đoković conjectures on permanents of doubly stochastic matrices, Linear Algebra Appl. 79 (1986), 127–151. doi:10.1016/0024-3795(86)90296-X
  5. D. K. Chang, Minimum and maximum permanents of certain doubly stochastic matrices, Linear and Multilinear Algebra 24 (1988), 39–44. doi:10.1080/03081088808817895
  6. T. H. Foregger, Permanents of convex combinations of doubly stochastic matrices, Linear and Multilinear Algebra 23 (1988), 79–90. doi:10.1080/03081088808817858
  7. S.-G. Hwang, Some permanental inequalities, Bull. Korean Math. Soc. 26 (1989), 35–42.
  8. S.-G. Hwang, On the monotonicity of the permanent, Proc. Amer. Math. Soc. 106 (1989), 59–63. doi:10.1090/S0002-9939-1989-0960645-2
  9. Y. Lavi, The Marcus–Minc transform inequality, preprint (2026). arXiv:2609.29262
  10. Y. Lavi, The Wang transform inequality, preprint (2026). Paper page