Second Immanantal Polynomial

• The second immanantal polynomial is a hook immanant defined via the partition (2,1^(n-2)), bridging determinant and permanent frameworks.
• It uses explicit character formulas and fixed-point statistics to provide concrete combinatorial interpretations in graph and Laplacian matrices.
• Its structural recurrences and complexity distinctions, depending on hook conventions, reveal key insights into reconstruction and computational tractability.

The second immanantal polynomial is an immanantal polynomial attached to a hook partition of $n$, most commonly $(2,1^{n-2})$ in the graph, Laplacian, and recent hook-immanant literature, where one writes

$$d_2(M)=\sum_{\sigma\in S_n}\chi_2(\sigma)\prod_{i=1}^n m_{i,\sigma(i)}$$

for the corresponding immanant and $d_2(xI-M)$ for the polynomial. The terminology is not completely uniform. In some representation-theoretic, hook-inequality, and Newton-polytope discussions, the adjacent hook $(n-1,1)$ is treated as the “second” immanant instead. Complexity-theoretic work on partition-indexed immanants often avoids the phrase entirely and works directly with $\mathrm{imm}_\lambda$ for a partition $\lambda\vdash n$. The resulting ambiguity is substantive rather than cosmetic, because the two hooks are conjugate but support different explicit character formulas, graph-theoretic interpretations, and complexity behavior (Nagar et al., 2017, Wu et al., 18 Feb 2025, Nagar, 2023, Curticapean, 2021).

1. Terminology and defining formulas

For a partition $\lambda\vdash n$ and an $n\times n$ matrix $A=(a_{ij})$, the immanant is

$(2,1^{n-2})$0

and the corresponding immanantal polynomial is

$(2,1^{n-2})$1

For $(2,1^{n-2})$2, this reduces to the characteristic polynomial, while for general $(2,1^{n-2})$3 it defines a partition-indexed family interpolating between determinant and permanent (Nagar et al., 2017).

In the usage that is explicit in several graph-matrix papers, the second immanant is the hook case $(2,1^{n-2})$4, and the second immanantal polynomial of a matrix $(2,1^{n-2})$5 is $(2,1^{n-2})$6. This convention is stated directly for general matrices, adjacency matrices, and linear-combination graph matrices, and it is the convention adopted in recent reconstruction and hook-immanant polynomial work (Wu et al., 18 Feb 2025, Dong et al., 25 Aug 2025, Dong et al., 6 Apr 2026).

A different convention appears in some literature organized from the permanent side of the hook chain. There the natural “second” object is the hook $(2,1^{n-2})$7, so that the relevant polynomial is $(2,1^{n-2})$8 in Laplacian settings, or $(2,1^{n-2})$9 in character-immanant settings (Nagar, 2023, Skandera, 30 Sep 2025). This terminological split is one of the central points in any precise account of the subject.

2. Character-theoretic placement within the immanant family

Immanants sit between the determinant and permanent. For the sign character one has

$$d_2(M)=\sum_{\sigma\in S_n}\chi_2(\sigma)\prod_{i=1}^n m_{i,\sigma(i)}$$0

and for the trivial character one has

$$d_2(M)=\sum_{\sigma\in S_n}\chi_2(\sigma)\prod_{i=1}^n m_{i,\sigma(i)}$$1

Thus the hook cases immediately adjacent to these extremes are $$d_2(M)=\sum_{\sigma\in S_n}\chi_2(\sigma)\prod_{i=1}^n m_{i,\sigma(i)}$$2 and $$d_2(M)=\sum_{\sigma\in S_n}\chi_2(\sigma)\prod_{i=1}^n m_{i,\sigma(i)}$$3, which are transposes of each other: $$d_2(M)=\sum_{\sigma\in S_n}\chi_2(\sigma)\prod_{i=1}^n m_{i,\sigma(i)}$$4 This transpose relation is frequently the source of the competing “second” conventions (Curticapean, 2021).

For the determinant-side hook $$d_2(M)=\sum_{\sigma\in S_n}\chi_2(\sigma)\prod_{i=1}^n m_{i,\sigma(i)}$$5, a particularly useful explicit formula is

$$d_2(M)=\sum_{\sigma\in S_n}\chi_2(\sigma)\prod_{i=1}^n m_{i,\sigma(i)}$$6

hence

$$d_2(M)=\sum_{\sigma\in S_n}\chi_2(\sigma)\prod_{i=1}^n m_{i,\sigma(i)}$$7

This gives the second immanant, in the $$d_2(M)=\sum_{\sigma\in S_n}\chi_2(\sigma)\prod_{i=1}^n m_{i,\sigma(i)}$$8 convention, as a sign-weighted deformation of the determinant by a fixed-point factor (Curticapean, 2021).

For the permanent-side hook $$d_2(M)=\sum_{\sigma\in S_n}\chi_2(\sigma)\prod_{i=1}^n m_{i,\sigma(i)}$$9, the corresponding character is

$d_2(xI-M)$0

In that convention the immanant is

$d_2(xI-M)$1

so the sign factor disappears and the weighting is governed directly by fixed points (Zhang, 8 Apr 2026).

3. Laplacian second immanantal polynomials of trees

For a tree $d_2(xI-M)$2 on $d_2(xI-M)$3 vertices with Laplacian matrix $d_2(xI-M)$4, the Laplacian immanantal polynomial is

$d_2(xI-M)$5

The $d_2(xI-M)$6-Laplacian is

$d_2(xI-M)$7

with $d_2(xI-M)$8 when $d_2(xI-M)$9, and the corresponding $(n-1,1)$0-Laplacian immanantal polynomial is

$(n-1,1)$1

In this setting, the phrase “second immanantal polynomial” is used explicitly for the partition

$(n-1,1)$2

following Merris’s usage (Nagar et al., 2017).

A central theorem states that if $(n-1,1)$3 covers $(n-1,1)$4 in the generalized tree shift poset $(n-1,1)$5, then for every partition $(n-1,1)$6 and every $(n-1,1)$7,

$(n-1,1)$8

At $(n-1,1)$9, this yields coefficientwise monotonicity for ordinary Laplacian immanantal polynomials. Specializing to $\mathrm{imm}_\lambda$0 shows that every coefficient of the second Laplacian immanantal polynomial decreases as one moves upward in $\mathrm{imm}_\lambda$1 (Nagar et al., 2017).

The second case is distinguished by an especially simple coefficient formula. If $\mathrm{imm}_\lambda$2 denotes the orientation statistic built from $\mathrm{imm}_\lambda$3-vertex orientations, then

$\mathrm{imm}_\lambda$4

For $\mathrm{imm}_\lambda$5, the coefficient acquires a moment interpretation: $\mathrm{imm}_\lambda$6 At $\mathrm{imm}_\lambda$7, this recovers Merris’s theorem relating the second Laplacian immanantal polynomial to the sum of vertex moments and, consequently, to centroid-type structure (Nagar et al., 2017).

4. Coefficients for graph linear-combination matrices

A broad recent framework studies immanantal polynomials of

$\mathrm{imm}_\lambda$8

where $\mathrm{imm}_\lambda$9 is the degree matrix and $\lambda\vdash n$0 the adjacency matrix of a graph $\lambda\vdash n$1. For hook partitions $\lambda\vdash n$2, the hook immanantal polynomial is

$\lambda\vdash n$3

Within this family, $\lambda\vdash n$4 is explicitly identified as the second immanant polynomial, while $\lambda\vdash n$5 is the characteristic polynomial and $\lambda\vdash n$6 the permanental polynomial (Dong et al., 25 Aug 2025).

For the second immanantal polynomial of the linear-combination matrix $\lambda\vdash n$7,

$\lambda\vdash n$8

the first coefficients are

$\lambda\vdash n$9

$\lambda\vdash n$0

and

$\lambda\vdash n$1

For the Laplacian specialization $\lambda\vdash n$2, these become

$\lambda\vdash n$3

$\lambda\vdash n$4

$\lambda\vdash n$5

and

$\lambda\vdash n$6

These formulas place the second immanantal polynomial in direct contact with degree symmetric sums, matching statistics, and cycle counts (Dong et al., 6 Apr 2026).

The same framework yields structural equalities and recursions. In particular, for bipartite graphs,

$\lambda\vdash n$7

where $\lambda\vdash n$8 is the signless Laplacian. More generally, the $\lambda\vdash n$9 specialization of the hook recursions expresses $n\times n$0 in terms of deleted vertices, deleted edges, and, for graphs with cycles, deleted cycle-vertex sets; for trees the cycle terms vanish, leaving especially tractable recurrences (Dong et al., 25 Aug 2025).

5. Complexity-theoretic interpretation and the importance of convention

The complexity of “the” second immanantal polynomial depends decisively on which hook partition is meant. For a partition $n\times n$1, let

$n\times n$2

the number of boxes to the right of the first column of the Young diagram. For a family $n\times n$3, define $n\times n$4, and let $n\times n$5 denote the problem of evaluating $n\times n$6 on input $n\times n$7 and $n\times n$8 (Curticapean, 2021).

If one interprets the second immanant as the determinant-side hook $n\times n$9, then $A=(a_{ij})$0 and

$A=(a_{ij})$1

For the family $A=(a_{ij})$2, one has $A=(a_{ij})$3, so

$A=(a_{ij})$4

In this interpretation, the second immanant lies on the polynomial-time side of the dichotomy (Curticapean, 2021).

If instead one interprets the second immanant as the permanent-side hook $A=(a_{ij})$5, then $A=(a_{ij})$6 and

$A=(a_{ij})$7

For the family $A=(a_{ij})$8, $A=(a_{ij})$9, indeed linearly growing. The dichotomy then yields much stronger hardness: this family is $(2,1^{n-2})$00-hard and $(2,1^{n-2})$01-complete, and it admits no $(2,1^{n-2})$02-time algorithm unless $(2,1^{n-2})$03 fails (Curticapean, 2021).

The complexity paper itself does not use the phrase “second immanantal polynomial”; the specialization to either hook is an inference from partition indexing. That inference, however, is exact enough to show that the terminology is computationally nontrivial: one natural “second” hook is easy, the other is hard (Curticapean, 2021).

6. Reconstruction, inequalities, and later generalizations

In reconstruction theory, the second immanantal polynomial in the $(2,1^{n-2})$04 convention satisfies deletion identities analogous to those known for characteristic and permanental polynomials. For a simple graph $(2,1^{n-2})$05 with

$(2,1^{n-2})$06

one has

$(2,1^{n-2})$07

Hence, when $(2,1^{n-2})$08, $(2,1^{n-2})$09 can be reconstructed from the second immanantal polynomials of the graphs in

$(2,1^{n-2})$10

For a digraph $(2,1^{n-2})$11, the analogous identity is

$(2,1^{n-2})$12

which yields reconstruction of

$(2,1^{n-2})$13

from the deleted-edge deck whenever $(2,1^{n-2})$14 (Wu et al., 18 Feb 2025).

Under the alternate $(2,1^{n-2})$15 convention, a different line of work studies normalized hook immanants on totally nonnegative matrices. There the principal theorem gives the hook chain

$(2,1^{n-2})$16

for every totally nonnegative matrix $(2,1^{n-2})$17. Since $(2,1^{n-2})$18, the normalized $(2,1^{n-2})$19-immanant lies immediately below the permanent and above the later hook immanants, and the difference

$(2,1^{n-2})$20

is itself totally nonnegative (Skandera, 30 Sep 2025).

Several recent generalizations move beyond the classical symmetric-function setting. Quasi-immanants replace cycle type by cycle composition and symmetric functions by quasisymmetric functions. The quasi-immanant attached to the quasisymmetric Schur function $(2,1^{n-2})$21 is presented as an analogue of the second immanant rather than as the classical second immanant itself; its coefficients depend on the first part of $(2,1^{n-2})$22 and vanish when $(2,1^{n-2})$23 (Campbell, 26 Jan 2025). In another direction, Newton-polytope work treats $(2,1^{n-2})$24 as the relevant second immanant: for Jacobi–Trudi matrices $(2,1^{n-2})$25 is SNP when $(2,1^{n-2})$26 is a border strip, and for Giambelli matrices $(2,1^{n-2})$27 is SNP for every $(2,1^{n-2})$28, hence in particular for $(2,1^{n-2})$29 (Zhang, 8 Apr 2026).

The cumulative picture is therefore bifurcated but coherent. In graph-theoretic and Laplacian applications, the second immanantal polynomial is usually $(2,1^{n-2})$30; in some hook-chain, representation-theoretic, and Newton-polytope settings, the label shifts to $(2,1^{n-2})$31. The two conventions are linked by conjugation of partitions, yet they lead to different explicit formulas, different monotonicity and positivity statements, and, in families, different computational complexity classes.